3.20.66 \(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=363 \[ \frac {\left (-2 c e \left (-d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

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Rubi [A]  time = 1.29, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {738, 826, 1166, 208} \begin {gather*} \frac {\left (-2 c e \left (-d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a + b*x + c*x^2)^2,x]

[Out]

-((Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + ((8*c^2*d^2 + b*(b - Sq
rt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sq
rt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*
c])*e]) - ((8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(
Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*
Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 208

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

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 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 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 {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{-b^2+4 a c}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {1}{2} d e (2 c d-b e)+\frac {1}{2} e \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2-4 a c}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]  time = 2.87, size = 418, normalized size = 1.15 \begin {gather*} \frac {\frac {1}{2} \left (e (a e-b d)+c d^2\right ) \left (\frac {\sqrt {2} \left (\frac {\left (2 c e \left (d \sqrt {b^2-4 a c}+2 a e-4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}}+4 e \sqrt {d+e x}\right )+\frac {(d+e x)^{5/2} \left (-2 c (a e+c d x)+b^2 e+b c (e x-d)\right )}{a+x (b+c x)}+e (d+e x)^{3/2} (2 c d-b e)}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c*d - b*e)*(d + e*x)^(3/2) + ((d + e*x)^(5/2)*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b +
c*x)) + ((c*d^2 + e*(-(b*d) + a*e))*(4*e*Sqrt[d + e*x] + (Sqrt[2]*(((8*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2
+ 2*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 -
2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c])))/2)/((b^2 - 4*a*c)*(c*
d^2 + e*(-(b*d) + a*e)))

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IntegrateAlgebraic [C]  time = 4.39, size = 545, normalized size = 1.50 \begin {gather*} \frac {\left (-2 \sqrt {2} c d e \sqrt {4 a c-b^2}+\sqrt {2} b e^2 \sqrt {4 a c-b^2}+4 i \sqrt {2} a c e^2+i \sqrt {2} b^2 e^2-8 i \sqrt {2} b c d e+8 i \sqrt {2} c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-i e \sqrt {4 a c-b^2}+b e-2 c d}}+\frac {\left (-2 \sqrt {2} c d e \sqrt {4 a c-b^2}+\sqrt {2} b e^2 \sqrt {4 a c-b^2}-4 i \sqrt {2} a c e^2-i \sqrt {2} b^2 e^2+8 i \sqrt {2} b c d e-8 i \sqrt {2} c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {i e \sqrt {4 a c-b^2}+b e-2 c d}}+\frac {e \sqrt {d+e x} \left (2 a e^2+b e (d+e x)-2 b d e+2 c d^2-2 c d (d+e x)\right )}{\left (b^2-4 a c\right ) \left (a e^2+b e (d+e x)-b d e+c d^2-2 c d (d+e x)+c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*Sqrt[d + e*x]*(2*c*d^2 - 2*b*d*e + 2*a*e^2 - 2*c*d*(d + e*x) + b*e*(d + e*x)))/((b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2 - 2*c*d*(d + e*x) + b*e*(d + e*x) + c*(d + e*x)^2)) + (((8*I)*Sqrt[2]*c^2*d^2 - (8*I)*Sqrt[2]*b*c*d*
e - 2*Sqrt[2]*c*Sqrt[-b^2 + 4*a*c]*d*e + I*Sqrt[2]*b^2*e^2 + (4*I)*Sqrt[2]*a*c*e^2 + Sqrt[2]*b*Sqrt[-b^2 + 4*a
*c]*e^2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(2*Sqrt[c]*(b^2
- 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) + (((-8*I)*Sqrt[2]*c^2*d^2 + (8*I)*Sq
rt[2]*b*c*d*e - 2*Sqrt[2]*c*Sqrt[-b^2 + 4*a*c]*d*e - I*Sqrt[2]*b^2*e^2 - (4*I)*Sqrt[2]*a*c*e^2 + Sqrt[2]*b*Sqr
t[-b^2 + 4*a*c]*e^2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(2*S
qrt[c]*(b^2 - 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e])

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fricas [B]  time = 0.49, size = 2396, normalized size = 6.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e
+ 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*s
qrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a
^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c
^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2
*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c
)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b
^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt
((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 4
8*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^
4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2
- 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d -
(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*
a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2
*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)
*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e
^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 -
64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c
^2)*e^4 - 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a
^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*
b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64
*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2
*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2
- 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^
2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*
a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2
)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 - 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c
^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a
*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6
*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt
(e*x + d)) - 2*(b*d - 2*a*e + (2*c*d - b*e)*x)*sqrt(e*x + d))/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3
- 4*a*b*c)*x)

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giac [B]  time = 1.28, size = 1223, normalized size = 3.37

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(2*(x*e + d)^(3/2)*c*d*e - 2*sqrt(x*e + d)*c*d^2*e - (x*e + d)^(3/2)*b*e^2 + 2*sqrt(x*e + d)*b*d*e^2 - 2*sqrt
(x*e + d)*a*e^3)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)*(b^2 - 4*a*c)) - 1
/8*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2*c*d*e - b*e^2) + 4*(sqrt(b^2 - 4*a
*c)*c^2*d^2*e - sqrt(b^2 - 4*a*c)*b*c*d*e^2 + sqrt(b^2 - 4*a*c)*a*c*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4
*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 2*(5*b^4
*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^3 - (b^5 - 16*a^2*b*c^2)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c
)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt((2*b^2*c*d - 8*
a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)
*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b*c^2)*sqr
t(b^2 - 4*a*c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 1/8*(sqrt(-4*
c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*(2*c*d*e - b*e^2) - 4*(sqrt(b^2 - 4*a*c)*c^2*d^2*
e - sqrt(b^2 - 4*a*c)*b*c*d*e^2 + sqrt(b^2 - 4*a*c)*a*c*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*
abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + 2*(5*b^4*c - 16*a*b^
2*c^2 - 16*a^2*c^3)*d*e^3 - (b^5 - 16*a^2*b*c^2)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan
(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e - sqrt((2*b^2*c*d - 8*a*c^2*d - b^
3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*(b^2*c - 4*
a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*
c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c))

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maple [B]  time = 0.13, size = 1503, normalized size = 4.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x)

[Out]

-e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*b+2*e/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(
3/2)*c*d-2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*a+2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^
2)*(e*x+d)^(1/2)*b*d-2*e/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*c*d^2-2*e^3/(4*a*c-b^2)*c/(-(4*a*
c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e
+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-1/2*e^3/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c
*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^
(1/2)*c)*b^2+4*e^2/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1
/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d-4*e/(4*a*c-b^2)*c^2/(
-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/
((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2+1/2*e^2/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*
e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b-e/(4*a
*c-b^2)*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(
-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d-2*e^3/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-
1/2*e^3/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*
x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2+4*e^2/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2
)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d-4*e/(4*a*c-b^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-
b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2-1
/2*e^2/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2
*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b+e/(4*a*c-b^2)*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^
(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a)^2, x)

________________________________________________________________________________________

mupad [B]  time = 6.32, size = 5326, normalized size = 14.67

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a + b*x + c*x^2)^2,x)

[Out]

log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d
^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - (2^(1/2)*((2^(1/2)*(8*c^2*e^3*(a*e^2 + c*d^2 - b*
d*e) - 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2)
+ 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2
*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*
e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b
^2)^6))^(1/2))*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d
^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 5
12*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*
a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/4 + (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 +
32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*
c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 3
2*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c
^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2
- 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/4)*(-(b^
9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e
^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 -
18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e -
576*a^2*b^4*c^3*d*e^2)/(8*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a
^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*
b^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - (2^(1/2)
*((2^(1/2)*(8*c^2*e^3*(a*e^2 + c*d^2 - b*d*e) - 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*
((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c
^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e
^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*
e + 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6
*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a
^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c
^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))/
4 + (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 + 32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2
*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*((e^3*(-(4*a*c - b^2)^9)^
(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e
^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576
*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2
)/(c*(4*a*c - b^2)^6))^(1/2))/4)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3
- 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^
2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*
c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(8*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^
2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) - ((2*(d + e*x)^(1/2)*(a*e^3 - b*d
*e^2 + c*d^2*e))/(4*a*c - b^2) + (e*(b*e - 2*c*d)*(d + e*x)^(3/2))/(4*a*c - b^2))/((b*e - 2*c*d)*(d + e*x) + c
*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b
^3*d*e^3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - ((8*c^2*e
^3*(a*e^2 + c*d^2 - b*d*e) + 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(((e^3*(-(4*a*c - b^2)^9)^(
1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*
e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72
*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c
*(4*a*c - b^2)^6))^(1/2))*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 -
48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*
c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2
*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2) - (2*c*e^2*(d + e*x)^(1/2)*(b^4*
e^4 + 32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10
*b^3*c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*
c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*
b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*
d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*(((e^3
*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4
*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (
9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*
a^2*b^4*c^3*d*e^2)/(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*
c^5 - 6144*a^5*b^2*c^6))^(1/2) - log((c*e^3*(b*e - 2*c*d)*(16*c^3*d^4 + 3*a*b^2*e^4 + 4*a^2*c*e^4 - 3*b^3*d*e^
3 + 20*a*c^2*d^2*e^2 + 19*b^2*c*d^2*e^2 - 32*b*c^2*d^3*e - 20*a*b*c*d*e^3))/(4*a*c - b^2)^3 - ((8*c^2*e^3*(a*e
^2 + c*d^2 - b*d*e) + 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b
^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 +
6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5
*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*
c - b^2)^6))^(1/2))*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a
*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e
^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e +
288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2) - (2*c*e^2*(d + e*x)^(1/2)*(b^4*e^4 +
32*c^4*d^4 + 8*a^2*c^2*e^4 + 24*a*c^3*d^2*e^2 + 42*b^2*c^2*d^2*e^2 + 2*a*b^2*c*e^4 - 64*b*c^3*d^3*e - 10*b^3*
c*d*e^3 - 24*a*b*c^2*d*e^3))/(4*a*c - b^2)^2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*
d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*
c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^
2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(c*(4*a*c - b^2)^6))^(1/2))*(-((b^9*e^
3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^
3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b
^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2
*b^4*c^3*d*e^2)/(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5
- 6144*a^5*b^2*c^6))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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