3.2379 \(\int \frac{1}{(d+e x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=519 \[ -\frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (64 a^2 e^2+332 a b d e+119 b^2 d^2\right )-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right )}{24 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right )}{12 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^2 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^3*Sqrt[a + b*x + c*x^2]) - (e*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*
e*(3*b*d + 4*a*e))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)^2*(d + e*x)^3) - (e*(2*c*d - b*e)*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(6*b*d + 2
9*a*e))*Sqrt[a + b*x + c*x^2])/(12*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)^2) - (e*(96*c^4*d^4 + 105*b^4*e^4 - 20*b^2*c*e^3*(19*b*d + 23*a*e) - 16*c^3
*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2 + 332*a*b*d*e + 64*a^2*e^2))*S
qrt[a + b*x + c*x^2])/(24*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (
5*e^2*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*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
])])/(16*(c*d^2 - b*d*e + a*e^2)^(9/2))

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Rubi [A]  time = 2.13339, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (64 a^2 e^2+332 a b d e+119 b^2 d^2\right )-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right )}{24 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right )}{12 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^2 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^3*Sqrt[a + b*x + c*x^2]) - (e*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*
e*(3*b*d + 4*a*e))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)^2*(d + e*x)^3) - (e*(2*c*d - b*e)*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(6*b*d + 2
9*a*e))*Sqrt[a + b*x + c*x^2])/(12*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x)^2) - (e*(96*c^4*d^4 + 105*b^4*e^4 - 20*b^2*c*e^3*(19*b*d + 23*a*e) - 16*c^3
*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2 + 332*a*b*d*e + 64*a^2*e^2))*S
qrt[a + b*x + c*x^2])/(24*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (
5*e^2*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*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
])])/(16*(c*d^2 - b*d*e + a*e^2)^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 7.39443, size = 680, normalized size = 1.31 \[ \frac{\left (a+b x+c x^2\right )^2 \left (\frac{2 \left (5 a^2 b c^2 e^4-8 a^2 c^3 d e^3+2 a^2 c^3 e^4 x-5 a b^3 c e^4+16 a b^2 c^2 d e^3-4 a b^2 c^2 e^4 x-18 a b c^3 d^2 e^2+12 a b c^3 d e^3 x+8 a c^4 d^3 e-12 a c^4 d^2 e^2 x+b^5 e^4-4 b^4 c d e^3+b^4 c e^4 x+6 b^3 c^2 d^2 e^2-4 b^3 c^2 d e^3 x-4 b^2 c^3 d^3 e+6 b^2 c^3 d^2 e^2 x+b c^4 d^4-4 b c^4 d^3 e x+2 c^5 d^4 x\right )}{\left (4 a c-b^2\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^4}+\frac{e^3 \left (40 a c e^2-57 b^2 e^2+188 b c d e-188 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac{11 e^3 (b e-2 c d)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e^3}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}\right )}{(a+x (b+c x))^{3/2}}-\frac{5 e^2 \left (a+b x+c x^2\right )^{3/2} (b e-2 c d) \log (d+e x) \left (-12 a c e^2+7 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{16 (a+x (b+c x))^{3/2} \left (a e^2-b d e+c d^2\right )^{9/2}}+\frac{5 e^2 \left (a+b x+c x^2\right )^{3/2} (b e-2 c d) \left (-12 a c e^2+7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \log \left (2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{16 (a+x (b+c x))^{3/2} \left (a e^2-b d e+c d^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((a + b*x + c*x^2)^2*(-e^3/(3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) + (11*e^3*(
-2*c*d + b*e))/(12*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) + (e^3*(-188*c^2*d^2 +
188*b*c*d*e - 57*b^2*e^2 + 40*a*c*e^2))/(24*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)
) + (2*(b*c^4*d^4 - 4*b^2*c^3*d^3*e + 8*a*c^4*d^3*e + 6*b^3*c^2*d^2*e^2 - 18*a*b
*c^3*d^2*e^2 - 4*b^4*c*d*e^3 + 16*a*b^2*c^2*d*e^3 - 8*a^2*c^3*d*e^3 + b^5*e^4 -
5*a*b^3*c*e^4 + 5*a^2*b*c^2*e^4 + 2*c^5*d^4*x - 4*b*c^4*d^3*e*x + 6*b^2*c^3*d^2*
e^2*x - 12*a*c^4*d^2*e^2*x - 4*b^3*c^2*d*e^3*x + 12*a*b*c^3*d*e^3*x + b^4*c*e^4*
x - 4*a*b^2*c^2*e^4*x + 2*a^2*c^3*e^4*x))/((-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2
)^4*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) - (5*e^2*(-2*c*d + b*e)*(16*c^2
*d^2 - 16*b*c*d*e + 7*b^2*e^2 - 12*a*c*e^2)*(a + b*x + c*x^2)^(3/2)*Log[d + e*x]
)/(16*(c*d^2 - b*d*e + a*e^2)^(9/2)*(a + x*(b + c*x))^(3/2)) + (5*e^2*(-2*c*d +
b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2 - 12*a*c*e^2)*(a + b*x + c*x^2)^(3/2)*
Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*
x + c*x^2]])/(16*(c*d^2 - b*d*e + a*e^2)^(9/2)*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.035, size = 3823, normalized size = 7.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(c*x^2+b*x+a)^(3/2),x)

[Out]

-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)-35/16*e^4/(a*e^2-b*d*e+c*d^2)^4/(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3+4/3*c/(a*e^2-b*d*e+c*d^2)^2/(d/e+
x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+7/12/(a*e^2
-b*d*e+c*d^2)^2/(d/e+x)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)
/e^2)^(1/2)*b+35/2*e/(a*e^2-b*d*e+c*d^2)^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^3*d^3-35/2*e/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2))/(d/e+x))*c^3*d^3-15/2*e/(a*e^2-b*d*e+c*d^2)^3*c^2/(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*d+35/6*e/(a*e^2-b*d*e+c*d^2)^3/(d
/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d+10
5/4*e^2/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e
+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c^2*d^2*b+35/8*e
^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)*x*b^4*c+115/3*e/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)/(c*
(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d-35/2*e^3/(a
*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)*b^4*c*d+105/2*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c^2*d^2-105/8*e^3/(a
*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^
2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2*c*d-70*e/(a*e^2-b*d*e+
c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b^2*c^3*d^3-115/6*e^2/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)/(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2+105*e^2/(a*e^2-b*d*e+
c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*x*b^2*c^3*d^2+70/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^5*d^4-115/3/(a*e^2-b*d*e+c*d^2)^
3*c^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*b*d^2-115/12*e^2/(a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3-105/4*e^2/(a*e^2-b*d*e+c*d^2)^4/(
c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2*b-15/4*
e^2/(a*e^2-b*d*e+c*d^2)^3*c/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c
*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b+15/2*e/(a*e^2-b*
d*e+c*d^2)^3*c^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(
b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*d-7/6/e/(a*e^2-b*d*e+c*d^2)^2
/(d/e+x)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d
+105/8*e^3/(a*e^2-b*d*e+c*d^2)^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*b^2*c*d+35/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^4*d^4-230/3/(a*e^2-b*d*e+c
*d^2)^3*c^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*x*d^2+230/3*e/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*d-140*e/(a*e^2-b*d*e+c*d^2)^
4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
x*c^4*d^3*b-35*e^3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3*c^2*d+35/16*e^4/(a*e^2-b*d*e+c*d^2)
^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*b^5+35/16*e^4/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^
2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^3+32/3*
c^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*x+16/3*c^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-35/6/(a*e^2-b*d*e+c*d^2
)^3/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^
2*d^2-35/24*e^2/(a*e^2-b*d*e+c*d^2)^3/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+15/4*e^2/(a*e^2-b*d*e+c*d^2)^3*c/(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 19.5695, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

[-1/96*(4*(48*b*c^4*d^7 - 192*(b^2*c^3 - 2*a*c^4)*d^6*e + 288*(b^3*c^2 - 3*a*b*c
^3)*d^5*e^2 - 48*(4*b^4*c - 21*a*b^2*c^2 + 28*a^2*c^3)*d^4*e^3 + 6*(8*b^5 - 85*a
*b^3*c + 220*a^2*b*c^2)*d^3*e^4 + (87*a*b^4 - 328*a^2*b^2*c - 80*a^3*c^2)*d^2*e^
5 - 38*(a^2*b^3 - 4*a^3*b*c)*d*e^6 + 8*(a^3*b^2 - 4*a^4*c)*e^7 + (96*c^5*d^4*e^3
- 192*b*c^4*d^3*e^4 + 4*(119*b^2*c^3 - 332*a*c^4)*d^2*e^5 - 4*(95*b^3*c^2 - 332
*a*b*c^3)*d*e^6 + (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^7)*x^4 + (288*c^5*
d^5*e^2 - 528*b*c^4*d^4*e^3 + 84*(13*b^2*c^3 - 36*a*c^4)*d^3*e^4 - 2*(271*b^3*c^
2 - 940*a*b*c^3)*d^2*e^5 - 4*(25*b^4*c - 91*a*b^2*c^2 - 12*a^2*c^3)*d*e^6 + (105
*b^5 - 530*a*b^3*c + 488*a^2*b*c^2)*e^7)*x^3 + (288*c^5*d^6*e - 432*b*c^4*d^5*e^
2 + 48*(11*b^2*c^3 - 32*a*c^4)*d^4*e^3 + 6*(73*b^3*c^2 - 244*a*b*c^3)*d^3*e^4 -
(787*b^4*c - 3356*a*b^2*c^2 + 1696*a^2*c^3)*d^2*e^5 + 2*(140*b^5 - 763*a*b^3*c +
884*a^2*b*c^2)*d*e^6 + (35*a*b^4 - 172*a^2*b^2*c + 128*a^3*c^2)*e^7)*x^2 + (96*
c^5*d^7 - 48*b*c^4*d^6*e - 288*(b^2*c^3 - 2*a*c^4)*d^5*e^2 + 48*(19*b^3*c^2 - 62
*a*b*c^3)*d^4*e^3 - 6*(133*b^4*c - 602*a*b^2*c^2 + 456*a^2*c^3)*d^3*e^4 + (231*b
^5 - 1490*a*b^3*c + 2408*a^2*b*c^2)*d^2*e^5 + 2*(49*a*b^4 - 214*a^2*b^2*c + 72*a
^3*c^2)*d*e^6 - 14*(a^2*b^3 - 4*a^3*b*c)*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqr
t(c*x^2 + b*x + a) - 15*(32*(a*b^2*c^3 - 4*a^2*c^4)*d^6*e^2 - 48*(a*b^3*c^2 - 4*
a^2*b*c^3)*d^5*e^3 + 6*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16*a^3*c^3)*d^4*e^4 - (7*a*
b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*d^3*e^5 + (32*(b^2*c^4 - 4*a*c^5)*d^3*e^5 - 4
8*(b^3*c^3 - 4*a*b*c^4)*d^2*e^6 + 6*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d*e^
7 - (7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*e^8)*x^5 + (96*(b^2*c^4 - 4*a*c^5)*d
^4*e^4 - 112*(b^3*c^3 - 4*a*b*c^4)*d^3*e^5 + 6*(7*b^4*c^2 - 40*a*b^2*c^3 + 48*a^
2*c^4)*d^2*e^6 + 3*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*d*e^7 - (7*b^6 - 40*a*
b^4*c + 48*a^2*b^2*c^2)*e^8)*x^4 + (96*(b^2*c^4 - 4*a*c^5)*d^5*e^3 - 48*(b^3*c^3
- 4*a*b*c^4)*d^4*e^4 - 2*(27*b^4*c^2 - 88*a*b^2*c^3 - 80*a^2*c^4)*d^3*e^5 + 3*(
23*b^5*c - 120*a*b^3*c^2 + 112*a^2*b*c^3)*d^2*e^6 - 3*(7*b^6 - 50*a*b^4*c + 96*a
^2*b^2*c^2 - 32*a^3*c^3)*d*e^7 - (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*e^8)*x^
3 + (32*(b^2*c^4 - 4*a*c^5)*d^6*e^2 + 48*(b^3*c^3 - 4*a*b*c^4)*d^5*e^3 - 6*(19*b
^4*c^2 - 88*a*b^2*c^3 + 48*a^2*c^4)*d^4*e^4 + (83*b^5*c - 536*a*b^3*c^2 + 816*a^
2*b*c^3)*d^3*e^5 - 3*(7*b^6 - 70*a*b^4*c + 192*a^2*b^2*c^2 - 96*a^3*c^3)*d^2*e^6
- 3*(7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*d*e^7)*x^2 + (32*(b^3*c^3 - 4*a*b*c
^4)*d^6*e^2 - 48*(b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^5*e^3 + 6*(5*b^5*c - 48*a
*b^3*c^2 + 112*a^2*b*c^3)*d^4*e^4 - (7*b^6 - 130*a*b^4*c + 480*a^2*b^2*c^2 - 288
*a^3*c^3)*d^3*e^5 - 3*(7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*d^2*e^6)*x)*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 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*
d*e + a*e^2) - 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c
^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/
(e^2*x^2 + 2*d*e*x + d^2)))/(((a*b^2*c^4 - 4*a^2*c^5)*d^11 - 4*(a*b^3*c^3 - 4*a^
2*b*c^4)*d^10*e + 2*(3*a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^9*e^2 - 4*(a*b^
5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^8*e^3 + (a*b^6 + 8*a^2*b^4*c - 42*a^3*b^2*c^
2 - 24*a^4*c^3)*d^7*e^4 - 4*(a^2*b^5 - a^3*b^3*c - 12*a^4*b*c^2)*d^6*e^5 + 2*(3*
a^3*b^4 - 10*a^4*b^2*c - 8*a^5*c^2)*d^5*e^6 - 4*(a^4*b^3 - 4*a^5*b*c)*d^4*e^7 +
(a^5*b^2 - 4*a^6*c)*d^3*e^8 + ((b^2*c^5 - 4*a*c^6)*d^8*e^3 - 4*(b^3*c^4 - 4*a*b*
c^5)*d^7*e^4 + 2*(3*b^4*c^3 - 10*a*b^2*c^4 - 8*a^2*c^5)*d^6*e^5 - 4*(b^5*c^2 - a
*b^3*c^3 - 12*a^2*b*c^4)*d^5*e^6 + (b^6*c + 8*a*b^4*c^2 - 42*a^2*b^2*c^3 - 24*a^
3*c^4)*d^4*e^7 - 4*(a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^3*e^8 + 2*(3*a^2*b^4
*c - 10*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*e^9 - 4*(a^3*b^3*c - 4*a^4*b*c^2)*d*e^10 +
(a^4*b^2*c - 4*a^5*c^2)*e^11)*x^5 + (3*(b^2*c^5 - 4*a*c^6)*d^9*e^2 - 11*(b^3*c^4
- 4*a*b*c^5)*d^8*e^3 + 2*(7*b^4*c^3 - 22*a*b^2*c^4 - 24*a^2*c^5)*d^7*e^4 - 2*(3
*b^5*c^2 + 4*a*b^3*c^3 - 64*a^2*b*c^4)*d^6*e^5 - (b^6*c - 28*a*b^4*c^2 + 78*a^2*
b^2*c^3 + 72*a^3*c^4)*d^5*e^6 + (b^7 - 4*a*b^5*c - 30*a^2*b^3*c^2 + 120*a^3*b*c^
3)*d^4*e^7 - 2*(2*a*b^6 - 11*a^2*b^4*c + 6*a^3*b^2*c^2 + 24*a^4*c^3)*d^3*e^8 + 2
*(3*a^2*b^5 - 16*a^3*b^3*c + 16*a^4*b*c^2)*d^2*e^9 - (4*a^3*b^4 - 19*a^4*b^2*c +
12*a^5*c^2)*d*e^10 + (a^4*b^3 - 4*a^5*b*c)*e^11)*x^4 + (3*(b^2*c^5 - 4*a*c^6)*d
^10*e - 9*(b^3*c^4 - 4*a*b*c^5)*d^9*e^2 + (6*b^4*c^3 - 11*a*b^2*c^4 - 52*a^2*c^5
)*d^8*e^3 + 2*(3*b^5*c^2 - 26*a*b^3*c^3 + 56*a^2*b*c^4)*d^7*e^4 - (9*b^6*c - 42*
a*b^4*c^2 + 2*a^2*b^2*c^3 + 88*a^3*c^4)*d^6*e^5 + (3*b^7 + 8*a*b^5*c - 110*a^2*b
^3*c^2 + 120*a^3*b*c^3)*d^5*e^6 - (11*a*b^6 - 38*a^2*b^4*c - 42*a^3*b^2*c^2 + 72
*a^4*c^3)*d^4*e^7 + 2*(7*a^2*b^5 - 34*a^3*b^3*c + 24*a^4*b*c^2)*d^3*e^8 - (6*a^3
*b^4 - 31*a^4*b^2*c + 28*a^5*c^2)*d^2*e^9 - (a^4*b^3 - 4*a^5*b*c)*d*e^10 + (a^5*
b^2 - 4*a^6*c)*e^11)*x^3 + ((b^2*c^5 - 4*a*c^6)*d^11 - (b^3*c^4 - 4*a*b*c^5)*d^1
0*e - (6*b^4*c^3 - 31*a*b^2*c^4 + 28*a^2*c^5)*d^9*e^2 + 2*(7*b^5*c^2 - 34*a*b^3*
c^3 + 24*a^2*b*c^4)*d^8*e^3 - (11*b^6*c - 38*a*b^4*c^2 - 42*a^2*b^2*c^3 + 72*a^3
*c^4)*d^7*e^4 + (3*b^7 + 8*a*b^5*c - 110*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^6*e^5 -
(9*a*b^6 - 42*a^2*b^4*c + 2*a^3*b^2*c^2 + 88*a^4*c^3)*d^5*e^6 + 2*(3*a^2*b^5 - 2
6*a^3*b^3*c + 56*a^4*b*c^2)*d^4*e^7 + (6*a^3*b^4 - 11*a^4*b^2*c - 52*a^5*c^2)*d^
3*e^8 - 9*(a^4*b^3 - 4*a^5*b*c)*d^2*e^9 + 3*(a^5*b^2 - 4*a^6*c)*d*e^10)*x^2 + ((
b^3*c^4 - 4*a*b*c^5)*d^11 - (4*b^4*c^3 - 19*a*b^2*c^4 + 12*a^2*c^5)*d^10*e + 2*(
3*b^5*c^2 - 16*a*b^3*c^3 + 16*a^2*b*c^4)*d^9*e^2 - 2*(2*b^6*c - 11*a*b^4*c^2 + 6
*a^2*b^2*c^3 + 24*a^3*c^4)*d^8*e^3 + (b^7 - 4*a*b^5*c - 30*a^2*b^3*c^2 + 120*a^3
*b*c^3)*d^7*e^4 - (a*b^6 - 28*a^2*b^4*c + 78*a^3*b^2*c^2 + 72*a^4*c^3)*d^6*e^5 -
2*(3*a^2*b^5 + 4*a^3*b^3*c - 64*a^4*b*c^2)*d^5*e^6 + 2*(7*a^3*b^4 - 22*a^4*b^2*
c - 24*a^5*c^2)*d^4*e^7 - 11*(a^4*b^3 - 4*a^5*b*c)*d^3*e^8 + 3*(a^5*b^2 - 4*a^6*
c)*d^2*e^9)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/48*(2*(48*b*c^4*d^7 - 192*(b^2*c
^3 - 2*a*c^4)*d^6*e + 288*(b^3*c^2 - 3*a*b*c^3)*d^5*e^2 - 48*(4*b^4*c - 21*a*b^2
*c^2 + 28*a^2*c^3)*d^4*e^3 + 6*(8*b^5 - 85*a*b^3*c + 220*a^2*b*c^2)*d^3*e^4 + (8
7*a*b^4 - 328*a^2*b^2*c - 80*a^3*c^2)*d^2*e^5 - 38*(a^2*b^3 - 4*a^3*b*c)*d*e^6 +
8*(a^3*b^2 - 4*a^4*c)*e^7 + (96*c^5*d^4*e^3 - 192*b*c^4*d^3*e^4 + 4*(119*b^2*c^
3 - 332*a*c^4)*d^2*e^5 - 4*(95*b^3*c^2 - 332*a*b*c^3)*d*e^6 + (105*b^4*c - 460*a
*b^2*c^2 + 256*a^2*c^3)*e^7)*x^4 + (288*c^5*d^5*e^2 - 528*b*c^4*d^4*e^3 + 84*(13
*b^2*c^3 - 36*a*c^4)*d^3*e^4 - 2*(271*b^3*c^2 - 940*a*b*c^3)*d^2*e^5 - 4*(25*b^4
*c - 91*a*b^2*c^2 - 12*a^2*c^3)*d*e^6 + (105*b^5 - 530*a*b^3*c + 488*a^2*b*c^2)*
e^7)*x^3 + (288*c^5*d^6*e - 432*b*c^4*d^5*e^2 + 48*(11*b^2*c^3 - 32*a*c^4)*d^4*e
^3 + 6*(73*b^3*c^2 - 244*a*b*c^3)*d^3*e^4 - (787*b^4*c - 3356*a*b^2*c^2 + 1696*a
^2*c^3)*d^2*e^5 + 2*(140*b^5 - 763*a*b^3*c + 884*a^2*b*c^2)*d*e^6 + (35*a*b^4 -
172*a^2*b^2*c + 128*a^3*c^2)*e^7)*x^2 + (96*c^5*d^7 - 48*b*c^4*d^6*e - 288*(b^2*
c^3 - 2*a*c^4)*d^5*e^2 + 48*(19*b^3*c^2 - 62*a*b*c^3)*d^4*e^3 - 6*(133*b^4*c - 6
02*a*b^2*c^2 + 456*a^2*c^3)*d^3*e^4 + (231*b^5 - 1490*a*b^3*c + 2408*a^2*b*c^2)*
d^2*e^5 + 2*(49*a*b^4 - 214*a^2*b^2*c + 72*a^3*c^2)*d*e^6 - 14*(a^2*b^3 - 4*a^3*
b*c)*e^7)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 15*(32*(a*b^2*
c^3 - 4*a^2*c^4)*d^6*e^2 - 48*(a*b^3*c^2 - 4*a^2*b*c^3)*d^5*e^3 + 6*(5*a*b^4*c -
24*a^2*b^2*c^2 + 16*a^3*c^3)*d^4*e^4 - (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*
d^3*e^5 + (32*(b^2*c^4 - 4*a*c^5)*d^3*e^5 - 48*(b^3*c^3 - 4*a*b*c^4)*d^2*e^6 + 6
*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d*e^7 - (7*b^5*c - 40*a*b^3*c^2 + 48*a^
2*b*c^3)*e^8)*x^5 + (96*(b^2*c^4 - 4*a*c^5)*d^4*e^4 - 112*(b^3*c^3 - 4*a*b*c^4)*
d^3*e^5 + 6*(7*b^4*c^2 - 40*a*b^2*c^3 + 48*a^2*c^4)*d^2*e^6 + 3*(3*b^5*c - 8*a*b
^3*c^2 - 16*a^2*b*c^3)*d*e^7 - (7*b^6 - 40*a*b^4*c + 48*a^2*b^2*c^2)*e^8)*x^4 +
(96*(b^2*c^4 - 4*a*c^5)*d^5*e^3 - 48*(b^3*c^3 - 4*a*b*c^4)*d^4*e^4 - 2*(27*b^4*c
^2 - 88*a*b^2*c^3 - 80*a^2*c^4)*d^3*e^5 + 3*(23*b^5*c - 120*a*b^3*c^2 + 112*a^2*
b*c^3)*d^2*e^6 - 3*(7*b^6 - 50*a*b^4*c + 96*a^2*b^2*c^2 - 32*a^3*c^3)*d*e^7 - (7
*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*e^8)*x^3 + (32*(b^2*c^4 - 4*a*c^5)*d^6*e^2
+ 48*(b^3*c^3 - 4*a*b*c^4)*d^5*e^3 - 6*(19*b^4*c^2 - 88*a*b^2*c^3 + 48*a^2*c^4)
*d^4*e^4 + (83*b^5*c - 536*a*b^3*c^2 + 816*a^2*b*c^3)*d^3*e^5 - 3*(7*b^6 - 70*a*
b^4*c + 192*a^2*b^2*c^2 - 96*a^3*c^3)*d^2*e^6 - 3*(7*a*b^5 - 40*a^2*b^3*c + 48*a
^3*b*c^2)*d*e^7)*x^2 + (32*(b^3*c^3 - 4*a*b*c^4)*d^6*e^2 - 48*(b^4*c^2 - 6*a*b^2
*c^3 + 8*a^2*c^4)*d^5*e^3 + 6*(5*b^5*c - 48*a*b^3*c^2 + 112*a^2*b*c^3)*d^4*e^4 -
(7*b^6 - 130*a*b^4*c + 480*a^2*b^2*c^2 - 288*a^3*c^3)*d^3*e^5 - 3*(7*a*b^5 - 40
*a^2*b^3*c + 48*a^3*b*c^2)*d^2*e^6)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*
(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)))
)/(((a*b^2*c^4 - 4*a^2*c^5)*d^11 - 4*(a*b^3*c^3 - 4*a^2*b*c^4)*d^10*e + 2*(3*a*b
^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^9*e^2 - 4*(a*b^5*c - a^2*b^3*c^2 - 12*a^3
*b*c^3)*d^8*e^3 + (a*b^6 + 8*a^2*b^4*c - 42*a^3*b^2*c^2 - 24*a^4*c^3)*d^7*e^4 -
4*(a^2*b^5 - a^3*b^3*c - 12*a^4*b*c^2)*d^6*e^5 + 2*(3*a^3*b^4 - 10*a^4*b^2*c - 8
*a^5*c^2)*d^5*e^6 - 4*(a^4*b^3 - 4*a^5*b*c)*d^4*e^7 + (a^5*b^2 - 4*a^6*c)*d^3*e^
8 + ((b^2*c^5 - 4*a*c^6)*d^8*e^3 - 4*(b^3*c^4 - 4*a*b*c^5)*d^7*e^4 + 2*(3*b^4*c^
3 - 10*a*b^2*c^4 - 8*a^2*c^5)*d^6*e^5 - 4*(b^5*c^2 - a*b^3*c^3 - 12*a^2*b*c^4)*d
^5*e^6 + (b^6*c + 8*a*b^4*c^2 - 42*a^2*b^2*c^3 - 24*a^3*c^4)*d^4*e^7 - 4*(a*b^5*
c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^3*e^8 + 2*(3*a^2*b^4*c - 10*a^3*b^2*c^2 - 8*a^
4*c^3)*d^2*e^9 - 4*(a^3*b^3*c - 4*a^4*b*c^2)*d*e^10 + (a^4*b^2*c - 4*a^5*c^2)*e^
11)*x^5 + (3*(b^2*c^5 - 4*a*c^6)*d^9*e^2 - 11*(b^3*c^4 - 4*a*b*c^5)*d^8*e^3 + 2*
(7*b^4*c^3 - 22*a*b^2*c^4 - 24*a^2*c^5)*d^7*e^4 - 2*(3*b^5*c^2 + 4*a*b^3*c^3 - 6
4*a^2*b*c^4)*d^6*e^5 - (b^6*c - 28*a*b^4*c^2 + 78*a^2*b^2*c^3 + 72*a^3*c^4)*d^5*
e^6 + (b^7 - 4*a*b^5*c - 30*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^4*e^7 - 2*(2*a*b^6 -
11*a^2*b^4*c + 6*a^3*b^2*c^2 + 24*a^4*c^3)*d^3*e^8 + 2*(3*a^2*b^5 - 16*a^3*b^3*c
+ 16*a^4*b*c^2)*d^2*e^9 - (4*a^3*b^4 - 19*a^4*b^2*c + 12*a^5*c^2)*d*e^10 + (a^4
*b^3 - 4*a^5*b*c)*e^11)*x^4 + (3*(b^2*c^5 - 4*a*c^6)*d^10*e - 9*(b^3*c^4 - 4*a*b
*c^5)*d^9*e^2 + (6*b^4*c^3 - 11*a*b^2*c^4 - 52*a^2*c^5)*d^8*e^3 + 2*(3*b^5*c^2 -
26*a*b^3*c^3 + 56*a^2*b*c^4)*d^7*e^4 - (9*b^6*c - 42*a*b^4*c^2 + 2*a^2*b^2*c^3
+ 88*a^3*c^4)*d^6*e^5 + (3*b^7 + 8*a*b^5*c - 110*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^
5*e^6 - (11*a*b^6 - 38*a^2*b^4*c - 42*a^3*b^2*c^2 + 72*a^4*c^3)*d^4*e^7 + 2*(7*a
^2*b^5 - 34*a^3*b^3*c + 24*a^4*b*c^2)*d^3*e^8 - (6*a^3*b^4 - 31*a^4*b^2*c + 28*a
^5*c^2)*d^2*e^9 - (a^4*b^3 - 4*a^5*b*c)*d*e^10 + (a^5*b^2 - 4*a^6*c)*e^11)*x^3 +
((b^2*c^5 - 4*a*c^6)*d^11 - (b^3*c^4 - 4*a*b*c^5)*d^10*e - (6*b^4*c^3 - 31*a*b^
2*c^4 + 28*a^2*c^5)*d^9*e^2 + 2*(7*b^5*c^2 - 34*a*b^3*c^3 + 24*a^2*b*c^4)*d^8*e^
3 - (11*b^6*c - 38*a*b^4*c^2 - 42*a^2*b^2*c^3 + 72*a^3*c^4)*d^7*e^4 + (3*b^7 + 8
*a*b^5*c - 110*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^6*e^5 - (9*a*b^6 - 42*a^2*b^4*c +
2*a^3*b^2*c^2 + 88*a^4*c^3)*d^5*e^6 + 2*(3*a^2*b^5 - 26*a^3*b^3*c + 56*a^4*b*c^2
)*d^4*e^7 + (6*a^3*b^4 - 11*a^4*b^2*c - 52*a^5*c^2)*d^3*e^8 - 9*(a^4*b^3 - 4*a^5
*b*c)*d^2*e^9 + 3*(a^5*b^2 - 4*a^6*c)*d*e^10)*x^2 + ((b^3*c^4 - 4*a*b*c^5)*d^11
- (4*b^4*c^3 - 19*a*b^2*c^4 + 12*a^2*c^5)*d^10*e + 2*(3*b^5*c^2 - 16*a*b^3*c^3 +
16*a^2*b*c^4)*d^9*e^2 - 2*(2*b^6*c - 11*a*b^4*c^2 + 6*a^2*b^2*c^3 + 24*a^3*c^4)
*d^8*e^3 + (b^7 - 4*a*b^5*c - 30*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^7*e^4 - (a*b^6 -
28*a^2*b^4*c + 78*a^3*b^2*c^2 + 72*a^4*c^3)*d^6*e^5 - 2*(3*a^2*b^5 + 4*a^3*b^3*
c - 64*a^4*b*c^2)*d^5*e^6 + 2*(7*a^3*b^4 - 22*a^4*b^2*c - 24*a^5*c^2)*d^4*e^7 -
11*(a^4*b^3 - 4*a^5*b*c)*d^3*e^8 + 3*(a^5*b^2 - 4*a^6*c)*d^2*e^9)*x)*sqrt(-c*d^2
+ b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

Exception raised: TypeError