3.15.2 \(\int \frac {b+2 c x}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=423 \[ \frac {e^2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e-5 b^3 e^2+9 b^2 c d e\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2} \]

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Rubi [A]  time = 0.57, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {822, 806, 724, 206} \begin {gather*} \frac {e^2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {2 e \left (-c x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-4 b c \left (c d^2-4 a e^2\right )-20 a c^2 d e+9 b^2 c d e-5 b^3 e^2\right )}{3 \left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {e^3 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a +
b*x + c*x^2)^(3/2)) - (2*e*(9*b^2*c*d*e - 20*a*c^2*d*e - 5*b^3*e^2 - 4*b*c*(c*d^2 - 4*a*e^2) - c*(8*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*
x^2]) + (e^2*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + b*x + c*x^2])/(3*(b^2 -
4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (e^3*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(b*d
- 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)
^(7/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], 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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (b^2-4 a c\right ) e (4 c d-5 b e)-3 c \left (b^2-4 a c\right ) e^2 x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} \left (b^2-4 a c\right ) e^2 \left (28 b^2 c d e-80 a c^2 d e-15 b^3 e^2-4 b c \left (2 c d^2-13 a e^2\right )\right )+\frac {1}{2} c \left (b^2-4 a c\right ) e^2 \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e^2 (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (e^3 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e^2 (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (e^3 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e \left (9 b^2 c d e-20 a c^2 d e-5 b^3 e^2-4 b c \left (c d^2-4 a e^2\right )-c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e^2 (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {e^3 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 1.35, size = 384, normalized size = 0.91 \begin {gather*} \frac {2 \left (\frac {1}{4} e^2 \left (\frac {2 \sqrt {a+x (b+c x)} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}\right )+\frac {\left (b^2-4 a c\right ) (b e-c d+c e x)}{(d+e x) (a+x (b+c x))^{3/2}}+\frac {e \left (4 b c \left (c d (d-2 e x)-4 a e^2\right )+4 c^2 \left (a e (5 d-3 e x)+2 c d^2 x\right )+5 b^3 e^2+b^2 c e (5 e x-9 d)\right )}{(d+e x) \sqrt {a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}\right )}{3 \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[(b + 2*c*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(((b^2 - 4*a*c)*(-(c*d) + b*e + c*e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (e*(5*b^3*e^2 + b^2*c*e*(-9*d
+ 5*e*x) + 4*c^2*(2*c*d^2*x + a*e*(5*d - 3*e*x)) + 4*b*c*(-4*a*e^2 + c*d*(d - 2*e*x))))/((c*d^2 + e*(-(b*d) +
a*e))*(d + e*x)*Sqrt[a + x*(b + c*x)]) + (e^2*((2*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a
*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*
e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a
+ x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)))/4))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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IntegrateAlgebraic [F]  time = 180.06, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

$Aborted

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fricas [B]  time = 10.04, size = 7442, normalized size = 17.59

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b*c^2)*d^2*e^4 + (5*a^2*b^4 - 24*a^3*b
^2*c + 16*a^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*d^2*e^4 - 16*(b^3*c^3 - 4*a*b*c^4)*d*e^5 + (5*b^4*c^2 - 24*
a*b^2*c^3 + 16*a^2*c^4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^3*e^3 + 16*(b^3*c^3 - 4*a*b*c^4)*d^2*e^4 - (27*b^
4*c^2 - 104*a*b^2*c^3 - 16*a^2*c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3
- 4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2*(3*b^5*c + 8*a*b^3*c^2 - 80*a^2*b*c^
3)*d*e^5 + (5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 32*a^3*c^3)*e^6)*x^3 + (16*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^
4)*d^3*e^3 - 16*(b^5*c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*d*e^5 + 2*(
5*a*b^5 - 24*a^2*b^3*c + 16*a^3*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^
2*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b^5 - 32*a^2*b^3*c + 48*a^3*b*c^2)*d*e^5 + (5*a^2*b^4 - 24*a^3*b^2*c +
16*a^4*c^2)*e^6)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 -
8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*
d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(2*(b^2*c^4 -
4*a*c^5)*d^7 - 8*(b^3*c^3 - 3*a*b*c^4)*d^6*e + 4*(3*b^4*c^2 - 2*a*b^2*c^3 - 20*a^2*c^4)*d^5*e^2 - 8*(b^5*c + 5
*a*b^3*c^2 - 26*a^2*b*c^3)*d^4*e^3 + 2*(b^6 + 24*a*b^4*c - 90*a^2*b^2*c^2 - 8*a^3*c^3)*d^3*e^4 - (16*a*b^5 - 4
1*a^2*b^3*c - 52*a^3*b*c^2)*d^2*e^5 + (11*a^2*b^4 - 54*a^3*b^2*c + 56*a^4*c^2)*d*e^6 + 3*(a^3*b^3 - 4*a^4*b*c)
*e^7 - (16*c^6*d^5*e^2 - 40*b*c^5*d^4*e^3 + 2*(31*b^2*c^4 - 44*a*c^5)*d^3*e^4 - (53*b^3*c^3 - 132*a*b*c^4)*d^2
*e^5 + (15*b^4*c^2 - 14*a*b^2*c^3 - 104*a^2*c^4)*d*e^6 - (15*a*b^3*c^2 - 52*a^2*b*c^3)*e^7)*x^4 - 2*(8*c^6*d^6
*e - 8*b*c^5*d^5*e^2 - (11*b^2*c^4 - 4*a*c^5)*d^4*e^3 + 4*(11*b^3*c^3 - 24*a*b*c^4)*d^3*e^4 - 2*(24*b^4*c^2 -
73*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^5 + (15*b^5*c - 24*a*b^3*c^2 - 88*a^2*b*c^3)*d*e^6 - 3*(5*a*b^4*c - 19*a^2*b^2
*c^2 + 4*a^3*c^3)*e^7)*x^3 - 3*(8*b*c^5*d^6*e - 2*(11*b^2*c^4 - 12*a*c^5)*d^5*e^2 + 8*(3*b^3*c^3 - 7*a*b*c^4)*
d^4*e^3 - 4*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*e^4 - (11*b^5*c - 26*a*b^3*c^2 - 32*a^2*b*c^3)*d^2*e^5 + (
5*b^6 - 8*a*b^4*c - 18*a^2*b^2*c^2 - 56*a^3*c^3)*d*e^6 - (5*a*b^5 - 14*a^2*b^3*c - 16*a^3*b*c^2)*e^7)*x^2 - 2*
((5*b^2*c^4 + 4*a*c^5)*d^6*e - 4*(5*b^3*c^3 - 6*a*b*c^4)*d^5*e^2 + 2*(15*b^4*c^2 - 34*a*b^2*c^3 - 4*a^2*c^4)*d
^4*e^3 - 4*(5*b^5*c - 17*a*b^3*c^2 + 8*a^2*b*c^3)*d^3*e^4 + (5*b^6 - 33*a*b^4*c + 69*a^2*b^2*c^2 - 28*a^3*c^3)
*d^2*e^5 + (5*a*b^5 - 4*a^2*b^3*c - 56*a^3*b*c^2)*d*e^6 - 2*(5*a^2*b^4 - 21*a^3*b^2*c + 8*a^4*c^2)*e^7)*x)*sqr
t(c*x^2 + b*x + a))/((a^2*b^2*c^4 - 4*a^3*c^5)*d^9 - 4*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^8*e + 2*(3*a^2*b^4*c^2 -
10*a^3*b^2*c^3 - 8*a^4*c^4)*d^7*e^2 - 4*(a^2*b^5*c - a^3*b^3*c^2 - 12*a^4*b*c^3)*d^6*e^3 + (a^2*b^6 + 8*a^3*b^
4*c - 42*a^4*b^2*c^2 - 24*a^5*c^3)*d^5*e^4 - 4*(a^3*b^5 - a^4*b^3*c - 12*a^5*b*c^2)*d^4*e^5 + 2*(3*a^4*b^4 - 1
0*a^5*b^2*c - 8*a^6*c^2)*d^3*e^6 - 4*(a^5*b^3 - 4*a^6*b*c)*d^2*e^7 + (a^6*b^2 - 4*a^7*c)*d*e^8 + ((b^2*c^6 - 4
*a*c^7)*d^8*e - 4*(b^3*c^5 - 4*a*b*c^6)*d^7*e^2 + 2*(3*b^4*c^4 - 10*a*b^2*c^5 - 8*a^2*c^6)*d^6*e^3 - 4*(b^5*c^
3 - a*b^3*c^4 - 12*a^2*b*c^5)*d^5*e^4 + (b^6*c^2 + 8*a*b^4*c^3 - 42*a^2*b^2*c^4 - 24*a^3*c^5)*d^4*e^5 - 4*(a*b
^5*c^2 - a^2*b^3*c^3 - 12*a^3*b*c^4)*d^3*e^6 + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d^2*e^7 - 4*(a^3
*b^3*c^2 - 4*a^4*b*c^3)*d*e^8 + (a^4*b^2*c^2 - 4*a^5*c^3)*e^9)*x^5 + ((b^2*c^6 - 4*a*c^7)*d^9 - 2*(b^3*c^5 - 4
*a*b*c^6)*d^8*e - 2*(b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*d^7*e^2 + 4*(2*b^5*c^3 - 9*a*b^3*c^4 + 4*a^2*b*c^5)*d^
6*e^3 - (7*b^6*c^2 - 16*a*b^4*c^3 - 54*a^2*b^2*c^4 + 24*a^3*c^5)*d^5*e^4 + 2*(b^7*c + 6*a*b^5*c^2 - 40*a^2*b^3
*c^3)*d^4*e^5 - 2*(4*a*b^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^3*e^6 + 4*(3*a^2*b^5*c - 11*a^3*b
^3*c^2 - 4*a^4*b*c^3)*d^2*e^7 - (8*a^3*b^4*c - 33*a^4*b^2*c^2 + 4*a^5*c^3)*d*e^8 + 2*(a^4*b^3*c - 4*a^5*b*c^2)
*e^9)*x^4 + (2*(b^3*c^5 - 4*a*b*c^6)*d^9 - (7*b^4*c^4 - 30*a*b^2*c^5 + 8*a^2*c^6)*d^8*e + 8*(b^5*c^3 - 4*a*b^3
*c^4)*d^7*e^2 - 2*(b^6*c^2 - 20*a^2*b^2*c^4 + 16*a^3*c^5)*d^6*e^3 - 2*(b^7*c - 6*a*b^5*c^2 + 14*a^2*b^3*c^3 -
24*a^3*b*c^4)*d^5*e^4 + (b^8 + 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^4*e^5 - 4*(a*b^7 -
2*a^2*b^5*c - 4*a^3*b^3*c^2 - 16*a^4*b*c^3)*d^3*e^6 + 2*(3*a^2*b^6 - 8*a^3*b^4*c - 12*a^4*b^2*c^2 - 16*a^5*c^3
)*d^2*e^7 - 2*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d*e^8 + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*e^9)*x^3 +
((b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^9 - 2*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^8*e + 2*(3*b^6*c^2 - 8
*a*b^4*c^3 - 12*a^2*b^2*c^4 - 16*a^3*c^5)*d^7*e^2 - 4*(b^7*c - 2*a*b^5*c^2 - 4*a^2*b^3*c^3 - 16*a^3*b*c^4)*d^6
*e^3 + (b^8 + 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^5*e^4 - 2*(a*b^7 - 6*a^2*b^5*c + 14*
a^3*b^3*c^2 - 24*a^4*b*c^3)*d^4*e^5 - 2*(a^2*b^6 - 20*a^4*b^2*c^2 + 16*a^5*c^3)*d^3*e^6 + 8*(a^3*b^5 - 4*a^4*b
^3*c)*d^2*e^7 - (7*a^4*b^4 - 30*a^5*b^2*c + 8*a^6*c^2)*d*e^8 + 2*(a^5*b^3 - 4*a^6*b*c)*e^9)*x^2 + (2*(a*b^3*c^
4 - 4*a^2*b*c^5)*d^9 - (8*a*b^4*c^3 - 33*a^2*b^2*c^4 + 4*a^3*c^5)*d^8*e + 4*(3*a*b^5*c^2 - 11*a^2*b^3*c^3 - 4*
a^3*b*c^4)*d^7*e^2 - 2*(4*a*b^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^6*e^3 + 2*(a*b^7 + 6*a^2*b^5
*c - 40*a^3*b^3*c^2)*d^5*e^4 - (7*a^2*b^6 - 16*a^3*b^4*c - 54*a^4*b^2*c^2 + 24*a^5*c^3)*d^4*e^5 + 4*(2*a^3*b^5
- 9*a^4*b^3*c + 4*a^5*b*c^2)*d^3*e^6 - 2*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^2*e^7 - 2*(a^5*b^3 - 4*a^6*b*c
)*d*e^8 + (a^6*b^2 - 4*a^7*c)*e^9)*x), -1/6*(3*(16*(a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^3 - 16*(a^2*b^3*c - 4*a^3*b
*c^2)*d^2*e^4 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2)*d*e^5 + (16*(b^2*c^4 - 4*a*c^5)*d^2*e^4 - 16*(b^3*c^3
- 4*a*b*c^4)*d*e^5 + (5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e^6)*x^5 + (16*(b^2*c^4 - 4*a*c^5)*d^3*e^3 + 16*(
b^3*c^3 - 4*a*b*c^4)*d^2*e^4 - (27*b^4*c^2 - 104*a*b^2*c^3 - 16*a^2*c^4)*d*e^5 + 2*(5*b^5*c - 24*a*b^3*c^2 + 1
6*a^2*b*c^3)*e^6)*x^4 + (32*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 - 16*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 2
*(3*b^5*c + 8*a*b^3*c^2 - 80*a^2*b*c^3)*d*e^5 + (5*b^6 - 14*a*b^4*c - 32*a^2*b^2*c^2 + 32*a^3*c^3)*e^6)*x^3 +
(16*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^3*e^3 - 16*(b^5*c - 4*a*b^3*c^2)*d^2*e^4 + (5*b^6 - 46*a*b^4*c + 96*
a^2*b^2*c^2 + 32*a^3*c^3)*d*e^5 + 2*(5*a*b^5 - 24*a^2*b^3*c + 16*a^3*b*c^2)*e^6)*x^2 + (32*(a*b^3*c^2 - 4*a^2*
b*c^3)*d^3*e^3 - 16*(2*a*b^4*c - 9*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*e^4 + 2*(5*a*b^5 - 32*a^2*b^3*c + 48*a^3*b*c^2
)*d*e^5 + (5*a^2*b^4 - 24*a^3*b^2*c + 16*a^4*c^2)*e^6)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2
+ b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^
2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + 2*(2*(b^2*c^4 - 4*a*c^5)*d^7 - 8*(b^3*c^3 - 3
*a*b*c^4)*d^6*e + 4*(3*b^4*c^2 - 2*a*b^2*c^3 - 20*a^2*c^4)*d^5*e^2 - 8*(b^5*c + 5*a*b^3*c^2 - 26*a^2*b*c^3)*d^
4*e^3 + 2*(b^6 + 24*a*b^4*c - 90*a^2*b^2*c^2 - 8*a^3*c^3)*d^3*e^4 - (16*a*b^5 - 41*a^2*b^3*c - 52*a^3*b*c^2)*d
^2*e^5 + (11*a^2*b^4 - 54*a^3*b^2*c + 56*a^4*c^2)*d*e^6 + 3*(a^3*b^3 - 4*a^4*b*c)*e^7 - (16*c^6*d^5*e^2 - 40*b
*c^5*d^4*e^3 + 2*(31*b^2*c^4 - 44*a*c^5)*d^3*e^4 - (53*b^3*c^3 - 132*a*b*c^4)*d^2*e^5 + (15*b^4*c^2 - 14*a*b^2
*c^3 - 104*a^2*c^4)*d*e^6 - (15*a*b^3*c^2 - 52*a^2*b*c^3)*e^7)*x^4 - 2*(8*c^6*d^6*e - 8*b*c^5*d^5*e^2 - (11*b^
2*c^4 - 4*a*c^5)*d^4*e^3 + 4*(11*b^3*c^3 - 24*a*b*c^4)*d^3*e^4 - 2*(24*b^4*c^2 - 73*a*b^2*c^3 + 8*a^2*c^4)*d^2
*e^5 + (15*b^5*c - 24*a*b^3*c^2 - 88*a^2*b*c^3)*d*e^6 - 3*(5*a*b^4*c - 19*a^2*b^2*c^2 + 4*a^3*c^3)*e^7)*x^3 -
3*(8*b*c^5*d^6*e - 2*(11*b^2*c^4 - 12*a*c^5)*d^5*e^2 + 8*(3*b^3*c^3 - 7*a*b*c^4)*d^4*e^3 - 4*(b^4*c^2 - 6*a*b^
2*c^3 + 8*a^2*c^4)*d^3*e^4 - (11*b^5*c - 26*a*b^3*c^2 - 32*a^2*b*c^3)*d^2*e^5 + (5*b^6 - 8*a*b^4*c - 18*a^2*b^
2*c^2 - 56*a^3*c^3)*d*e^6 - (5*a*b^5 - 14*a^2*b^3*c - 16*a^3*b*c^2)*e^7)*x^2 - 2*((5*b^2*c^4 + 4*a*c^5)*d^6*e
- 4*(5*b^3*c^3 - 6*a*b*c^4)*d^5*e^2 + 2*(15*b^4*c^2 - 34*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - 4*(5*b^5*c - 17*a*b^
3*c^2 + 8*a^2*b*c^3)*d^3*e^4 + (5*b^6 - 33*a*b^4*c + 69*a^2*b^2*c^2 - 28*a^3*c^3)*d^2*e^5 + (5*a*b^5 - 4*a^2*b
^3*c - 56*a^3*b*c^2)*d*e^6 - 2*(5*a^2*b^4 - 21*a^3*b^2*c + 8*a^4*c^2)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^2
*c^4 - 4*a^3*c^5)*d^9 - 4*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^8*e + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d
^7*e^2 - 4*(a^2*b^5*c - a^3*b^3*c^2 - 12*a^4*b*c^3)*d^6*e^3 + (a^2*b^6 + 8*a^3*b^4*c - 42*a^4*b^2*c^2 - 24*a^5
*c^3)*d^5*e^4 - 4*(a^3*b^5 - a^4*b^3*c - 12*a^5*b*c^2)*d^4*e^5 + 2*(3*a^4*b^4 - 10*a^5*b^2*c - 8*a^6*c^2)*d^3*
e^6 - 4*(a^5*b^3 - 4*a^6*b*c)*d^2*e^7 + (a^6*b^2 - 4*a^7*c)*d*e^8 + ((b^2*c^6 - 4*a*c^7)*d^8*e - 4*(b^3*c^5 -
4*a*b*c^6)*d^7*e^2 + 2*(3*b^4*c^4 - 10*a*b^2*c^5 - 8*a^2*c^6)*d^6*e^3 - 4*(b^5*c^3 - a*b^3*c^4 - 12*a^2*b*c^5)
*d^5*e^4 + (b^6*c^2 + 8*a*b^4*c^3 - 42*a^2*b^2*c^4 - 24*a^3*c^5)*d^4*e^5 - 4*(a*b^5*c^2 - a^2*b^3*c^3 - 12*a^3
*b*c^4)*d^3*e^6 + 2*(3*a^2*b^4*c^2 - 10*a^3*b^2*c^3 - 8*a^4*c^4)*d^2*e^7 - 4*(a^3*b^3*c^2 - 4*a^4*b*c^3)*d*e^8
+ (a^4*b^2*c^2 - 4*a^5*c^3)*e^9)*x^5 + ((b^2*c^6 - 4*a*c^7)*d^9 - 2*(b^3*c^5 - 4*a*b*c^6)*d^8*e - 2*(b^4*c^4
- 6*a*b^2*c^5 + 8*a^2*c^6)*d^7*e^2 + 4*(2*b^5*c^3 - 9*a*b^3*c^4 + 4*a^2*b*c^5)*d^6*e^3 - (7*b^6*c^2 - 16*a*b^4
*c^3 - 54*a^2*b^2*c^4 + 24*a^3*c^5)*d^5*e^4 + 2*(b^7*c + 6*a*b^5*c^2 - 40*a^2*b^3*c^3)*d^4*e^5 - 2*(4*a*b^6*c
- 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^3*e^6 + 4*(3*a^2*b^5*c - 11*a^3*b^3*c^2 - 4*a^4*b*c^3)*d^2*e^7
- (8*a^3*b^4*c - 33*a^4*b^2*c^2 + 4*a^5*c^3)*d*e^8 + 2*(a^4*b^3*c - 4*a^5*b*c^2)*e^9)*x^4 + (2*(b^3*c^5 - 4*a
*b*c^6)*d^9 - (7*b^4*c^4 - 30*a*b^2*c^5 + 8*a^2*c^6)*d^8*e + 8*(b^5*c^3 - 4*a*b^3*c^4)*d^7*e^2 - 2*(b^6*c^2 -
20*a^2*b^2*c^4 + 16*a^3*c^5)*d^6*e^3 - 2*(b^7*c - 6*a*b^5*c^2 + 14*a^2*b^3*c^3 - 24*a^3*b*c^4)*d^5*e^4 + (b^8
+ 2*a*b^6*c - 18*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^4*e^5 - 4*(a*b^7 - 2*a^2*b^5*c - 4*a^3*b^3*c^2 -
16*a^4*b*c^3)*d^3*e^6 + 2*(3*a^2*b^6 - 8*a^3*b^4*c - 12*a^4*b^2*c^2 - 16*a^5*c^3)*d^2*e^7 - 2*(2*a^3*b^5 - 5*
a^4*b^3*c - 12*a^5*b*c^2)*d*e^8 + (a^4*b^4 - 2*a^5*b^2*c - 8*a^6*c^2)*e^9)*x^3 + ((b^4*c^4 - 2*a*b^2*c^5 - 8*a
^2*c^6)*d^9 - 2*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^8*e + 2*(3*b^6*c^2 - 8*a*b^4*c^3 - 12*a^2*b^2*c^4 -
16*a^3*c^5)*d^7*e^2 - 4*(b^7*c - 2*a*b^5*c^2 - 4*a^2*b^3*c^3 - 16*a^3*b*c^4)*d^6*e^3 + (b^8 + 2*a*b^6*c - 18*
a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 48*a^4*c^4)*d^5*e^4 - 2*(a*b^7 - 6*a^2*b^5*c + 14*a^3*b^3*c^2 - 24*a^4*b*c^3)*d
^4*e^5 - 2*(a^2*b^6 - 20*a^4*b^2*c^2 + 16*a^5*c^3)*d^3*e^6 + 8*(a^3*b^5 - 4*a^4*b^3*c)*d^2*e^7 - (7*a^4*b^4 -
30*a^5*b^2*c + 8*a^6*c^2)*d*e^8 + 2*(a^5*b^3 - 4*a^6*b*c)*e^9)*x^2 + (2*(a*b^3*c^4 - 4*a^2*b*c^5)*d^9 - (8*a*b
^4*c^3 - 33*a^2*b^2*c^4 + 4*a^3*c^5)*d^8*e + 4*(3*a*b^5*c^2 - 11*a^2*b^3*c^3 - 4*a^3*b*c^4)*d^7*e^2 - 2*(4*a*b
^6*c - 7*a^2*b^4*c^2 - 38*a^3*b^2*c^3 + 8*a^4*c^4)*d^6*e^3 + 2*(a*b^7 + 6*a^2*b^5*c - 40*a^3*b^3*c^2)*d^5*e^4
- (7*a^2*b^6 - 16*a^3*b^4*c - 54*a^4*b^2*c^2 + 24*a^5*c^3)*d^4*e^5 + 4*(2*a^3*b^5 - 9*a^4*b^3*c + 4*a^5*b*c^2)
*d^3*e^6 - 2*(a^4*b^4 - 6*a^5*b^2*c + 8*a^6*c^2)*d^2*e^7 - 2*(a^5*b^3 - 4*a^6*b*c)*d*e^8 + (a^6*b^2 - 4*a^7*c)
*e^9)*x)]

________________________________________________________________________________________

giac [B]  time = 3.18, size = 6103, normalized size = 14.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((32*sqrt(c*d^2 - b*d*e + a*e^2)*c^4*d^3*e^3 + 48*b^2*c^(5/2)*d^2*e^5*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2
- b*d*e + a*e^2)*sqrt(c))) - 192*a*c^(7/2)*d^2*e^5*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(
c))) - 48*sqrt(c*d^2 - b*d*e + a*e^2)*b*c^3*d^2*e^4 - 48*b^3*c^(3/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2
- b*d*e + a*e^2)*sqrt(c))) + 192*a*b*c^(5/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(
c))) + 76*sqrt(c*d^2 - b*d*e + a*e^2)*b^2*c^2*d*e^5 - 208*sqrt(c*d^2 - b*d*e + a*e^2)*a*c^3*d*e^5 + 15*b^4*sqr
t(c)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 72*a*b^2*c^(3/2)*e^7*log(abs(-2*c*d
+ b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 48*a^2*c^(5/2)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d
*e + a*e^2)*sqrt(c))) - 30*sqrt(c*d^2 - b*d*e + a*e^2)*b^3*c*e^6 + 104*sqrt(c*d^2 - b*d*e + a*e^2)*a*b*c^2*e^6
)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e + a*e^2)*b^2*c^(7/2)*d^6 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*a*c^(9/2)*d^6
- 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^3*c^(5/2)*d^5*e + 12*sqrt(c*d^2 - b*d*e + a*e^2)*a*b*c^(7/2)*d^5*e + 3*sqrt(
c*d^2 - b*d*e + a*e^2)*b^4*c^(3/2)*d^4*e^2 - 9*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c^(5/2)*d^4*e^2 - 12*sqrt(c*d
^2 - b*d*e + a*e^2)*a^2*c^(7/2)*d^4*e^2 - sqrt(c*d^2 - b*d*e + a*e^2)*b^5*sqrt(c)*d^3*e^3 - 2*sqrt(c*d^2 - b*d
*e + a*e^2)*a*b^3*c^(3/2)*d^3*e^3 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b*c^(5/2)*d^3*e^3 + 3*sqrt(c*d^2 - b*d*
e + a*e^2)*a*b^4*sqrt(c)*d^2*e^4 - 9*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^2*c^(3/2)*d^2*e^4 - 12*sqrt(c*d^2 - b*d
*e + a*e^2)*a^3*c^(5/2)*d^2*e^4 - 3*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^3*sqrt(c)*d*e^5 + 12*sqrt(c*d^2 - b*d*e
+ a*e^2)*a^3*b*c^(3/2)*d*e^5 + sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^2*sqrt(c)*e^6 - 4*sqrt(c*d^2 - b*d*e + a*e^2)
*a^4*c^(3/2)*e^6) + 2*((((4*(4*b^2*c^5*d^6*e^17*sgn(1/(x*e + d)) - 16*a*c^6*d^6*e^17*sgn(1/(x*e + d)) - 12*b^3
*c^4*d^5*e^18*sgn(1/(x*e + d)) + 48*a*b*c^5*d^5*e^18*sgn(1/(x*e + d)) + 31*b^4*c^3*d^4*e^19*sgn(1/(x*e + d)) -
188*a*b^2*c^4*d^4*e^19*sgn(1/(x*e + d)) + 256*a^2*c^5*d^4*e^19*sgn(1/(x*e + d)) - 42*b^5*c^2*d^3*e^20*sgn(1/(
x*e + d)) + 296*a*b^3*c^3*d^3*e^20*sgn(1/(x*e + d)) - 512*a^2*b*c^4*d^3*e^20*sgn(1/(x*e + d)) + 24*b^6*c*d^2*e
^21*sgn(1/(x*e + d)) - 162*a*b^4*c^2*d^2*e^21*sgn(1/(x*e + d)) + 204*a^2*b^2*c^3*d^2*e^21*sgn(1/(x*e + d)) + 2
40*a^3*c^4*d^2*e^21*sgn(1/(x*e + d)) - 5*b^7*d*e^22*sgn(1/(x*e + d)) + 22*a*b^5*c*d*e^22*sgn(1/(x*e + d)) + 52
*a^2*b^3*c^2*d*e^22*sgn(1/(x*e + d)) - 240*a^3*b*c^3*d*e^22*sgn(1/(x*e + d)) + 5*a*b^6*e^23*sgn(1/(x*e + d)) -
41*a^2*b^4*c*e^23*sgn(1/(x*e + d)) + 92*a^3*b^2*c^2*e^23*sgn(1/(x*e + d)) - 32*a^4*c^3*e^23*sgn(1/(x*e + d)))
/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x
*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d
^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 +
24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e
+ d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^
3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 2
4*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x
*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sg
n(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2) - 3*(2*b^4*c^3*d^
5*e^20*sgn(1/(x*e + d)) - 16*a*b^2*c^4*d^5*e^20*sgn(1/(x*e + d)) + 32*a^2*c^5*d^5*e^20*sgn(1/(x*e + d)) - 5*b^
5*c^2*d^4*e^21*sgn(1/(x*e + d)) + 40*a*b^3*c^3*d^4*e^21*sgn(1/(x*e + d)) - 80*a^2*b*c^4*d^4*e^21*sgn(1/(x*e +
d)) + 4*b^6*c*d^3*e^22*sgn(1/(x*e + d)) - 28*a*b^4*c^2*d^3*e^22*sgn(1/(x*e + d)) + 32*a^2*b^2*c^3*d^3*e^22*sgn
(1/(x*e + d)) + 64*a^3*c^4*d^3*e^22*sgn(1/(x*e + d)) - b^7*d^2*e^23*sgn(1/(x*e + d)) + 2*a*b^5*c*d^2*e^23*sgn(
1/(x*e + d)) + 32*a^2*b^3*c^2*d^2*e^23*sgn(1/(x*e + d)) - 96*a^3*b*c^3*d^2*e^23*sgn(1/(x*e + d)) + 2*a*b^6*d*e
^24*sgn(1/(x*e + d)) - 14*a^2*b^4*c*d*e^24*sgn(1/(x*e + d)) + 16*a^3*b^2*c^2*d*e^24*sgn(1/(x*e + d)) + 32*a^4*
c^3*d*e^24*sgn(1/(x*e + d)) - a^2*b^5*e^25*sgn(1/(x*e + d)) + 8*a^3*b^3*c*e^25*sgn(1/(x*e + d)) - 16*a^4*b*c^2
*e^25*sgn(1/(x*e + d)))*e^(-1)/((b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2
+ 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(
1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2
*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d
))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1
/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*
d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d)
)^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1
/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/
(x*e + d))^2)*(x*e + d)))*e^(-1)/(x*e + d) - 3*(16*b^2*c^5*d^5*e^16*sgn(1/(x*e + d)) - 64*a*c^6*d^5*e^16*sgn(1
/(x*e + d)) - 40*b^3*c^4*d^4*e^17*sgn(1/(x*e + d)) + 160*a*b*c^5*d^4*e^17*sgn(1/(x*e + d)) + 84*b^4*c^3*d^3*e^
18*sgn(1/(x*e + d)) - 512*a*b^2*c^4*d^3*e^18*sgn(1/(x*e + d)) + 704*a^2*c^5*d^3*e^18*sgn(1/(x*e + d)) - 86*b^5
*c^2*d^2*e^19*sgn(1/(x*e + d)) + 608*a*b^3*c^3*d^2*e^19*sgn(1/(x*e + d)) - 1056*a^2*b*c^4*d^2*e^19*sgn(1/(x*e
+ d)) + 36*b^6*c*d*e^20*sgn(1/(x*e + d)) - 260*a*b^4*c^2*d*e^20*sgn(1/(x*e + d)) + 432*a^2*b^2*c^3*d*e^20*sgn(
1/(x*e + d)) + 128*a^3*c^4*d*e^20*sgn(1/(x*e + d)) - 5*b^7*e^21*sgn(1/(x*e + d)) + 34*a*b^5*c*e^21*sgn(1/(x*e
+ d)) - 40*a^2*b^3*c^2*e^21*sgn(1/(x*e + d)) - 64*a^3*b*c^3*e^21*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*
e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e
^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 +
3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(
1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14
*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 +
3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1
/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*
e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2
*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))*e^(-1)/(x*e + d) + 6*(8*b^2*c^5*d^4*e^15*sgn
(1/(x*e + d)) - 32*a*c^6*d^4*e^15*sgn(1/(x*e + d)) - 16*b^3*c^4*d^3*e^16*sgn(1/(x*e + d)) + 64*a*b*c^5*d^3*e^1
6*sgn(1/(x*e + d)) + 29*b^4*c^3*d^2*e^17*sgn(1/(x*e + d)) - 184*a*b^2*c^4*d^2*e^17*sgn(1/(x*e + d)) + 272*a^2*
c^5*d^2*e^17*sgn(1/(x*e + d)) - 21*b^5*c^2*d*e^18*sgn(1/(x*e + d)) + 152*a*b^3*c^3*d*e^18*sgn(1/(x*e + d)) - 2
72*a^2*b*c^4*d*e^18*sgn(1/(x*e + d)) + 5*b^6*c*e^19*sgn(1/(x*e + d)) - 39*a*b^4*c^2*e^19*sgn(1/(x*e + d)) + 80
*a^2*b^2*c^3*e^19*sgn(1/(x*e + d)) - 16*a^3*c^4*e^19*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 -
8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x
*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4
*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))
^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e
+ d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*
e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^
2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(
x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(
1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))*e^(-1)/(x*e + d) - (16*b^2*c^5*d^3*e^14*sgn(1/(x*e + d))
- 64*a*c^6*d^3*e^14*sgn(1/(x*e + d)) - 24*b^3*c^4*d^2*e^15*sgn(1/(x*e + d)) + 96*a*b*c^5*d^2*e^15*sgn(1/(x*e
+ d)) + 38*b^4*c^3*d*e^16*sgn(1/(x*e + d)) - 256*a*b^2*c^4*d*e^16*sgn(1/(x*e + d)) + 416*a^2*c^5*d*e^16*sgn(1/
(x*e + d)) - 15*b^5*c^2*e^17*sgn(1/(x*e + d)) + 112*a*b^3*c^3*e^17*sgn(1/(x*e + d)) - 208*a^2*b*c^4*e^17*sgn(1
/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e
^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48
*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*
e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14
*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*
a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e
+ d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e
^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3
*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))/(c -
2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)^(3/2) - 3*(16*c^
2*d^2*e^6 - 16*b*c*d*e^7 + 5*b^2*e^8 - 4*a*c*e^8)*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*(sqrt(c
- 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e
^2 - b*d*e^3 + a*e^4)*e^(-1)/(x*e + d))))/((c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 -
b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7)*sqrt(c*d^2 - b*d*
e + a*e^2)*sgn(1/(x*e + d))))*e^(-2)

________________________________________________________________________________________

maple [B]  time = 0.11, size = 4855, normalized size = 11.48 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*e^2/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+2/e/(a*e^2-b*d
*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d-160/3/(a*e^2-b*d*e+c*d
^2)*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b+20/3*e^2/(a*e^2-b*
d*e+c*d^2)^2*c/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4+10*e^2/(a*e
^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c^2*d^2+
10*e^3/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d*b+10/3*e/(a
*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d*b+10/(a*e^2-b*d*e+c*
d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*c^2*d^2+80/(a*e^2-b*d
*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d^2+2/3*c/
(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-20/3/(a*e^2-b*d*e+c*d^2)
*c^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*b-2*c*e^2/(a*e^2-b*d*e+c*
d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-30*e^3/(a*e^2-b*d*e+c
*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c^2*d+60*e^2/(a*e^
2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^3*d^2-80*
e/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*
b^2*d-10*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)
*c^2*x*b^2*d-5/6*e^2/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b
^2-5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-1/(a*e^
2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b-10/3/(a*e^2-b*d*e+c
*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*d^2-2*c*e^2/(a*e^2-b*d*e+c*d^2)^
2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))-80/3/(a*e^2-b*d*e+c*d^2)
*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-10/3/(a*e^2-b*d*e+c*d
^2)*c/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2+5/6*e^2/(a*e^2-b*d*e+c
*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^4-10*e^2/(a*e^2-b*d*e+
c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2+5/2*e^4/(a*e^2-b*d*e+c*d^2)
^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4+5/2*e^4/(a*e^2-b*d*e+c*d^
2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2-4*c^2*e^2/(a*e^2-
b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b+8*c^3*e/(a*e^
2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d+4*c^2*e/(a*
e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d+20/(a*e^2
-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^3*x*b*d^2+160/
(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*
d^2-40/3/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)
*c^4*x*d^3-20*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*b*c^3*d^3+5/3*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^
2)/e^2)^(3/2)*c*x*b^3+40/3*e^2/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3-320/3/e/(a*e^2-b*d*e+c*d^2)^2*c^5/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d
/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^3-20/3/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(
x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c^3*d^3-5*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*
d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^3*c*d+30*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e
-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c^2*d^2-15*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)
^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c*d+160/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^
2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d-160/3/e/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a
*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^3+5*e^4/(a*e^2-b*d*e+c*d^2)^3/
(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3*c-40*e/(a*e^2-b*d*e+c*d^2)
^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^4*d^3+40/3/e/(a*e^2-b*d*e
+c*d^2)*c^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*d+20/3/e/(a*e^2-b*
d*e+c*d^2)*c^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*d+320/3/e/(a*e^
2-b*d*e+c*d^2)*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-10*e^3/
(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d*
b-40*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*b^3*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
for more details)Is a*e^2-b*d*e                            +c*d^2    positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {b+2\,c\,x}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((2*c*x+b)/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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