3.2301 \(\int \frac {1}{x^{5/2} (a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=360 \[ \frac {b \left (5 b^2-19 a c\right )}{a^3 \sqrt {x} \left (b^2-4 a c\right )}-\frac {5 b^2-14 a c}{3 a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

1/3*(14*a*c-5*b^2)/a^2/(-4*a*c+b^2)/x^(3/2)+(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/x^(3/2)/(c*x^2+b*x+a)+b*(-19*a*c+
5*b^2)/a^3/(-4*a*c+b^2)/x^(1/2)+1/2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^
4-29*a*b^2*c+28*a^2*c^2+b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b-(-4*a*c+b^2)^(
1/2))^(1/2)-1/2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*
c^2-b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]  time = 3.69, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {740, 828, 826, 1166, 205} \[ \frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {5 b^2-14 a c}{3 a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \sqrt {x} \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^{3/2} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

-(5*b^2 - 14*a*c)/(3*a^2*(b^2 - 4*a*c)*x^(3/2)) + (b*(5*b^2 - 19*a*c))/(a^3*(b^2 - 4*a*c)*Sqrt[x]) + (b^2 - 2*
a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^(3/2)*(a + b*x + c*x^2)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^
2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^3*(b^
2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)
*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^3*(b^2 - 4*a*c)^
(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \left (a+b x+c x^2\right )^2} \, dx &=\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-5 b^2+14 a c\right )-\frac {5 b c x}{2}}{x^{5/2} \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} b \left (5 b^2-19 a c\right )+\frac {1}{2} c \left (5 b^2-14 a c\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-5 b^4+24 a b^2 c-14 a^2 c^2\right )-\frac {1}{2} b c \left (5 b^2-19 a c\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (-5 b^4+24 a b^2 c-14 a^2 c^2\right )-\frac {1}{2} b c \left (5 b^2-19 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}-\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {5 b^2-14 a c}{3 a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {b \left (5 b^2-19 a c\right )}{a^3 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^{3/2} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 0.90, size = 337, normalized size = 0.94 \[ \frac {\frac {b \left (5 b^2-19 a c\right )}{a^2 \sqrt {x}}-\frac {\sqrt {c} \left (\frac {\left (28 a^2 c^2-29 a b^2 c+19 a b c \sqrt {b^2-4 a c}-5 b^3 \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}-\frac {\left (28 a^2 c^2-29 a b^2 c-19 a b c \sqrt {b^2-4 a c}+5 b^3 \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c}}+\frac {-2 a c+b^2+b c x}{x^{3/2} (a+x (b+c x))}+\frac {14 a c-5 b^2}{3 a x^{3/2}}}{a \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

((-5*b^2 + 14*a*c)/(3*a*x^(3/2)) + (b*(5*b^2 - 19*a*c))/(a^2*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(x^(3/2)*(a + x*
(b + c*x))) - (Sqrt[c]*(-(((5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*
a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((5*b^4 -
29*a*b^2*c + 28*a^2*c^2 - 5*b^3*Sqrt[b^2 - 4*a*c] + 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x
])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]))/(a*(b^2 - 4*a*
c))

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fricas [B]  time = 2.56, size = 3449, normalized size = 9.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/6*(3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-(25
*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b
^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c
^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 -
12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 -
75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^11
- 94*a^8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8
250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(
a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2
415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 82
50*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a
^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*
c^3)) + 2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) - 3
*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-(25*b^9 -
315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2
- 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 2
4108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^
8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(-sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579
*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^11 - 94*
a^8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a
*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*
b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a
^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*
b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b
^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))
+ 2*(1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) + 3*sqrt
(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-(25*b^9 - 315*
a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64
*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*
a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4
*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log(sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b
^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^
9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*
c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 -
12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3
*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c
+ 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 1
2*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)) + 2*(
1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) - 3*sqrt(1/2)*
((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-(25*b^9 - 315*a*b^7*
c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*
c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^
2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 4
8*a^9*b^2*c^2 - 64*a^10*c^3))*log(-sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^
3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^9*c +
700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 3
9525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^
15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3
+ 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39
525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^1
5*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)) + 2*(1125*
b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*sqrt(x)) - 2*(2*a^2*b^2 - 8*
a^3*c - 3*(5*b^3*c - 19*a*b*c^2)*x^3 - (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*x^2 - 10*(a*b^3 - 4*a^2*b*c)*x)*sqrt
(x))/((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)

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giac [B]  time = 0.82, size = 3656, normalized size = 10.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

(b^3*c*x^(3/2) - 3*a*b*c^2*x^(3/2) + b^4*sqrt(x) - 4*a*b^2*c*sqrt(x) + 2*a^2*c^2*sqrt(x))/((a^3*b^2 - 4*a^4*c)
*(c*x^2 + b*x + a)) + 1/8*(10*a^6*b^9*c^2 - 138*a^7*b^7*c^3 + 680*a^8*b^5*c^4 - 1376*a^9*b^3*c^5 + 896*a^10*b*
c^6 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^9 + 69*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^8*c
- 340*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^2 - 98*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7
*c^2 + 688*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^3*c^3 + 288*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c^3 + 49*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a^7*b^5*c^3 - 448*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b*c^4 - 224*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^2*c^4 - 144*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^8*b^3*c^4 + 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b*c^5 - 10*(b^2 - 4*a*c
)*a^6*b^7*c^2 + 98*(b^2 - 4*a*c)*a^7*b^5*c^3 - 288*(b^2 - 4*a*c)*a^8*b^3*c^4 + 224*(b^2 - 4*a*c)*a^9*b*c^5 + (
10*b^5*c^2 - 78*a*b^3*c^3 + 152*a^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 +
39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*b^4*c - 76*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 38*sqrt
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*b^3*c^2 + 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 10*(b^2 - 4*a
*c)*b^3*c^2 + 38*(b^2 - 4*a*c)*a*b*c^3)*(a^3*b^2 - 4*a^4*c)^2 + 2*(5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^3*b^8 - 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3
*b^7*c - 10*a^3*b^8*c + 286*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^2 + 88*sqrt(2)*sqrt(b*c + sqrt(b
^2 - 4*a*c)*c)*a^4*b^5*c^2 + 5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^2 + 128*a^4*b^6*c^2 - 496*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^3 - 220*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^3 - 4
4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^3 - 572*a^5*b^4*c^3 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^7*c^4 + 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^4 + 110*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^5*b^2*c^4 + 992*a^6*b^2*c^4 - 56*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*c^5 - 448*a^7*c^5 + 10*
(b^2 - 4*a*c)*a^3*b^6*c - 88*(b^2 - 4*a*c)*a^4*b^4*c^2 + 220*(b^2 - 4*a*c)*a^5*b^2*c^3 - 112*(b^2 - 4*a*c)*a^6
*c^4)*abs(a^3*b^2 - 4*a^4*c))*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^3*b^3 - 4*a^4*b*c + sqrt((a^3*b^3 - 4*a^4*b*c
)^2 - 4*(a^4*b^2 - 4*a^5*c)*(a^3*b^2*c - 4*a^4*c^2)))/(a^3*b^2*c - 4*a^4*c^2)))/((a^7*b^6 - 12*a^8*b^4*c - 2*a
^7*b^5*c + 48*a^9*b^2*c^2 + 16*a^8*b^3*c^2 + a^7*b^4*c^2 - 64*a^10*c^3 - 32*a^9*b*c^3 - 8*a^8*b^2*c^3 + 16*a^9
*c^4)*abs(a^3*b^2 - 4*a^4*c)*abs(c)) - 1/8*(10*a^6*b^9*c^2 - 138*a^7*b^7*c^3 + 680*a^8*b^5*c^4 - 1376*a^9*b^3*
c^5 + 896*a^10*b*c^6 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^9 + 69*sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^6*b^8*c - 340*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^2 - 98*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*a^6*b^7*c^2 + 688*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^3*c^3 + 288*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c^3 + 49*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^7*b^5*c^3 - 448*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b*c^4 - 224*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^2*c^4 - 144*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a^8*b^3*c^4 + 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b*c^5
- 10*(b^2 - 4*a*c)*a^6*b^7*c^2 + 98*(b^2 - 4*a*c)*a^7*b^5*c^3 - 288*(b^2 - 4*a*c)*a^8*b^3*c^4 + 224*(b^2 - 4*a
*c)*a^9*b*c^5 + (10*b^5*c^2 - 78*a*b^3*c^3 + 152*a^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*b^5 + 39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 10*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c - 76*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
2*b*c^2 - 38*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^
3 - 10*(b^2 - 4*a*c)*b^3*c^2 + 38*(b^2 - 4*a*c)*a*b*c^3)*(a^3*b^2 - 4*a^4*c)^2 - 2*(5*sqrt(2)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a^3*b^8 - 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c - 10*sqrt(2)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*a^3*b^7*c + 10*a^3*b^8*c + 286*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^2 + 88*sqrt(2)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^2 + 5*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^2 - 128*a^4*
b^6*c^2 - 496*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^3 - 220*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c
)*a^5*b^3*c^3 - 44*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^3 + 572*a^5*b^4*c^3 + 224*sqrt(2)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^7*c^4 + 112*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b*c^4 + 110*sqrt(2)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^4 - 992*a^6*b^2*c^4 - 56*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*c^5 +
448*a^7*c^5 - 10*(b^2 - 4*a*c)*a^3*b^6*c + 88*(b^2 - 4*a*c)*a^4*b^4*c^2 - 220*(b^2 - 4*a*c)*a^5*b^2*c^3 + 112*
(b^2 - 4*a*c)*a^6*c^4)*abs(a^3*b^2 - 4*a^4*c))*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^3*b^3 - 4*a^4*b*c - sqrt((a^
3*b^3 - 4*a^4*b*c)^2 - 4*(a^4*b^2 - 4*a^5*c)*(a^3*b^2*c - 4*a^4*c^2)))/(a^3*b^2*c - 4*a^4*c^2)))/((a^7*b^6 - 1
2*a^8*b^4*c - 2*a^7*b^5*c + 48*a^9*b^2*c^2 + 16*a^8*b^3*c^2 + a^7*b^4*c^2 - 64*a^10*c^3 - 32*a^9*b*c^3 - 8*a^8
*b^2*c^3 + 16*a^9*c^4)*abs(a^3*b^2 - 4*a^4*c)*abs(c)) + 2/3*(6*b*x - a)/(a^3*x^(3/2))

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maple [B]  time = 0.13, size = 929, normalized size = 2.58 \[ \frac {14 \sqrt {2}\, c^{3} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a}+\frac {14 \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a}-\frac {29 \sqrt {2}\, b^{2} c^{2} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}-\frac {29 \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {5 \sqrt {2}\, b^{4} c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}+\frac {5 \sqrt {2}\, b^{4} c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}+\frac {3 b \,c^{2} x^{\frac {3}{2}}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{3} c \,x^{\frac {3}{2}}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {19 \sqrt {2}\, b \,c^{2} \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {19 \sqrt {2}\, b \,c^{2} \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2}}+\frac {5 \sqrt {2}\, b^{3} c \arctanh \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}-\frac {5 \sqrt {2}\, b^{3} c \arctan \left (\frac {\sqrt {2}\, c \sqrt {x}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{3}}-\frac {2 c^{2} \sqrt {x}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}+\frac {4 b^{2} c \sqrt {x}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{4} \sqrt {x}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}+\frac {4 b}{a^{3} \sqrt {x}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(c*x^2+b*x+a)^2,x)

[Out]

3/a^2/(c*x^2+b*x+a)*b*c^2/(4*a*c-b^2)*x^(3/2)-1/a^3/(c*x^2+b*x+a)*b^3*c/(4*a*c-b^2)*x^(3/2)-2/a/(c*x^2+b*x+a)/
(4*a*c-b^2)*x^(1/2)*c^2+4/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(1/2)*b^2*c-1/a^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x^(1/2)*
b^4+19/2/a^2/(4*a*c-b^2)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2
)^(1/2))*c)^(1/2))*b-5/2/a^3/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/(
(b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3+14/a/(4*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-29/2/a^2/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2+5/2/a
^3/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*
a*c+b^2)^(1/2))*c)^(1/2))*b^4-19/2/a^2/(4*a*c-b^2)*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(
1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b+5/2/a^3/(4*a*c-b^2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1
/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3+14/a/(4*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2
^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-29/2/a^2
/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(
-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2+5/2/a^3/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^
(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4-2/3/a^2/x^(3/2)+4/a^3*b/x^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 \, {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 14 \, a^{2} c^{3}\right )} x^{\frac {5}{2}} + 3 \, {\left (5 \, b^{5} - 19 \, a b^{3} c - 5 \, a^{2} b c^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (15 \, a b^{4} - 67 \, a^{2} b^{2} c + 28 \, a^{3} c^{2}\right )} \sqrt {x} + \frac {10 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )}}{\sqrt {x}} - \frac {2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )}}{x^{\frac {3}{2}}}}{3 \, {\left (a^{5} b^{2} - 4 \, a^{6} c + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{2} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )}} + \int -\frac {{\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 14 \, a^{2} c^{3}\right )} x^{\frac {3}{2}} + {\left (5 \, b^{5} - 29 \, a b^{3} c + 33 \, a^{2} b c^{2}\right )} \sqrt {x}}{2 \, {\left (a^{5} b^{2} - 4 \, a^{6} c + {\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{2} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

1/3*(3*(5*b^4*c - 24*a*b^2*c^2 + 14*a^2*c^3)*x^(5/2) + 3*(5*b^5 - 19*a*b^3*c - 5*a^2*b*c^2)*x^(3/2) + 2*(15*a*
b^4 - 67*a^2*b^2*c + 28*a^3*c^2)*sqrt(x) + 10*(a^2*b^3 - 4*a^3*b*c)/sqrt(x) - 2*(a^3*b^2 - 4*a^4*c)/x^(3/2))/(
a^5*b^2 - 4*a^6*c + (a^4*b^2*c - 4*a^5*c^2)*x^2 + (a^4*b^3 - 4*a^5*b*c)*x) + integrate(-1/2*((5*b^4*c - 24*a*b
^2*c^2 + 14*a^2*c^3)*x^(3/2) + (5*b^5 - 29*a*b^3*c + 33*a^2*b*c^2)*sqrt(x))/(a^5*b^2 - 4*a^6*c + (a^4*b^2*c -
4*a^5*c^2)*x^2 + (a^4*b^3 - 4*a^5*b*c)*x), x)

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mupad [B]  time = 4.04, size = 8768, normalized size = 24.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x + c*x^2)^2),x)

[Out]

atan((((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3
+ 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b
^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a
^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)
*(x^(1/2)*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c
^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*
a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 409
6*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1
/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a
^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 57344*a^19*c^9 + 20*a^12*b^14*c^2 - 496*a^13*b^12*c^3 + 5176*a^14*b^10*c^4
- 29280*a^15*b^8*c^5 + 96000*a^16*b^6*c^6 - 179200*a^17*b^4*c^7 + 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*
c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 +
300160*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 +
6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*
c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c
- b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^1
1*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*1i + ((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 +
6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*
c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c
- b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^1
1*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(57344*a^19*c^9 + x^(1/2)*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2)
- 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a
^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 1
65*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^1
0*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11
*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 20*a^12*b^14*c^2 +
496*a^13*b^12*c^3 - 5176*a^14*b^10*c^4 + 29280*a^15*b^8*c^5 - 96000*a^16*b^6*c^6 + 179200*a^17*b^4*c^7 - 16998
4*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 6101
2*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 300160*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*
c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*
b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c -
b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*
b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*1i)/(((-(25*b^15 - 25*b^6*(-(4*a*
c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*
b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c -
b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*
b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(x^(1/2)*(-(25*b^15 - 25*b^6*(-(4
*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a
^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*
c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a
^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^1
3*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7)
- 57344*a^19*c^9 + 20*a^12*b^14*c^2 - 496*a^13*b^12*c^3 + 5176*a^14*b^10*c^4 - 29280*a^15*b^8*c^5 + 96000*a^16
*b^6*c^6 - 179200*a^17*b^4*c^7 + 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*c^10 - 50*a^9*b^14*c^3 + 1180*a^10
*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 300160*a^14*b^4*c^8 - 233984*a^15
*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*
c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615
*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 40
96*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(
1/2) - ((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3
+ 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*
b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*
a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2
)*(57344*a^19*c^9 + x^(1/2)*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2
- 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^
2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/
(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*
a^12*b^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18
*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 20*a^12*b^14*c^2 + 496*a^13*b^12*c^3 - 5176*a^14*b^10*c^
4 + 29280*a^15*b^8*c^5 - 96000*a^16*b^6*c^6 + 179200*a^17*b^4*c^7 - 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16
*c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7
+ 300160*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 +
6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3
*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c
- b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^
11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2) + 119168*a^13*b*c^10 + 450*a^9*b^9*c^6 - 7270*a^10*b^7*c^7 + 44008*a^1
1*b^5*c^8 - 118304*a^12*b^3*c^9))*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^
11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*
c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(
1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 -
6144*a^12*b^2*c^5)))^(1/2)*2i - (2/(3*a) - (10*b*x)/(3*a^2) + (x^2*(15*b^4 + 14*a^2*c^2 - 62*a*b^2*c))/(3*a^3
*(4*a*c - b^2)) + (c*x^3*(5*b^3 - 19*a*b*c))/(a^3*(4*a*c - b^2)))/(a*x^(3/2) + b*x^(5/2) + c*x^(7/2)) + atan((
((-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 1169
28*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c
+ 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^
6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(x^(1
/2)*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 1
16928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13
*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13
*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(3
2768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a^19*b^
5*c^6 - 49152*a^20*b^3*c^7) - 57344*a^19*c^9 + 20*a^12*b^14*c^2 - 496*a^13*b^12*c^3 + 5176*a^14*b^10*c^4 - 292
80*a^15*b^8*c^5 + 96000*a^16*b^6*c^6 - 179200*a^17*b^4*c^7 + 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*c^10 -
50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 30016
0*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a
^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-
(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)
^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*
c^4 - 6144*a^12*b^2*c^5)))^(1/2)*1i + ((-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a
^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-
(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)
^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*
c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(57344*a^19*c^9 + x^(1/2)*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 8064
0*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3
*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b
^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*
c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 +
1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 20*a^12*b^14*c^2 + 496*a^
13*b^12*c^3 - 5176*a^14*b^10*c^4 + 29280*a^15*b^8*c^5 - 96000*a^16*b^6*c^6 + 179200*a^17*b^4*c^7 - 169984*a^18
*b^2*c^8) - x^(1/2)*(50176*a^16*c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12
*b^8*c^6 - 182336*a^13*b^6*c^7 + 300160*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 + 25*b^6*(-(4*a*c - b^
2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^
5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^
9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^
2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*1i)/(((-(25*b^15 + 25*b^6*(-(4*a*c - b^
2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^
5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^
9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^
2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(x^(1/2)*(-(25*b^15 + 25*b^6*(-(4*a*c -
b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5
*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^
2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8
*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2
- 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 5734
4*a^19*c^9 + 20*a^12*b^14*c^2 - 496*a^13*b^12*c^3 + 5176*a^14*b^10*c^4 - 29280*a^15*b^8*c^5 + 96000*a^16*b^6*c
^6 - 179200*a^17*b^4*c^7 + 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*c^10 - 50*a^9*b^14*c^3 + 1180*a^10*b^12*
c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 300160*a^14*b^4*c^8 - 233984*a^15*b^2*c
^9))*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 +
116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^1
3*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^1
3*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2) -
((-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116
928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c
+ 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c
^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(573
44*a^19*c^9 + x^(1/2)*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 357
67*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^2)^9)^
(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^
7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b
^2*c^5)))^(1/2)*(32768*a^21*b*c^8 + 8*a^15*b^13*c^2 - 192*a^16*b^11*c^3 + 1920*a^17*b^9*c^4 - 10240*a^18*b^7*c
^5 + 30720*a^19*b^5*c^6 - 49152*a^20*b^3*c^7) - 20*a^12*b^14*c^2 + 496*a^13*b^12*c^3 - 5176*a^14*b^10*c^4 + 29
280*a^15*b^8*c^5 - 96000*a^16*b^6*c^6 + 179200*a^17*b^4*c^7 - 169984*a^18*b^2*c^8) - x^(1/2)*(50176*a^16*c^10
- 50*a^9*b^14*c^3 + 1180*a^10*b^12*c^4 - 11602*a^11*b^10*c^5 + 61012*a^12*b^8*c^6 - 182336*a^13*b^6*c^7 + 3001
60*a^14*b^4*c^8 - 233984*a^15*b^2*c^9))*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*
a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(
-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2
)^9)^(1/2))/(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4
*c^4 - 6144*a^12*b^2*c^5)))^(1/2) + 119168*a^13*b*c^10 + 450*a^9*b^9*c^6 - 7270*a^10*b^7*c^7 + 44008*a^11*b^5*
c^8 - 118304*a^12*b^3*c^9))*(-(25*b^15 + 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2
- 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 - 49*a^3*c^3*(-(4*a*c - b^
2)^9)^(1/2) - 615*a*b^13*c + 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/
(8*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*
a^12*b^2*c^5)))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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