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

Optimal. Leaf size=430 \[ -\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (-35 a^3 e^6+47 a^2 c d^2 e^4+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5} \]

[Out]

(e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/(16*a^3*(c*d^
2 + a*e^2)^4*(d + e*x)) + (a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^
2)^3) - (a*e*(c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)/(24*a^2*(c*d^2 + a*
e^2)^2*(d + e*x)*(a + c*x^2)^2) - (a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3
*c*d*(5*c^2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(48*a^3*(c*d^2 + a*e^2)^3*(d +
e*x)*(a + c*x^2)) + (Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4
+ 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*(c*d
^2 + a*e^2)^5) + (8*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (4*c*d*e^7*Log[a +
c*x^2])/(c*d^2 + a*e^2)^5

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Rubi [A]  time = 1.45024, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (-35 a^3 e^6+47 a^2 c d^2 e^4+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

(e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/(16*a^3*(c*d^
2 + a*e^2)^4*(d + e*x)) + (a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^
2)^3) - (a*e*(c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)/(24*a^2*(c*d^2 + a*
e^2)^2*(d + e*x)*(a + c*x^2)^2) - (a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3
*c*d*(5*c^2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(48*a^3*(c*d^2 + a*e^2)^3*(d +
e*x)*(a + c*x^2)) + (Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4
+ 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*(c*d
^2 + a*e^2)^5) + (8*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (4*c*d*e^7*Log[a +
c*x^2])/(c*d^2 + a*e^2)^5

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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)**2/(c*x**2+a)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.853628, size = 336, normalized size = 0.78 \[ \frac{\frac{2 c \left (a e^2+c d^2\right )^2 \left (a^2 e^3 (24 d-11 e x)+18 a c d^2 e^2 x+5 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 c \left (a e^2+c d^2\right ) \left (a^3 e^5 (48 d-19 e x)+47 a^2 c d^2 e^4 x+23 a c^2 d^4 e^2 x+5 c^3 d^6 x\right )}{a^3 \left (a+c x^2\right )}+\frac{3 \sqrt{c} \left (-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{8 c \left (a e^2+c d^2\right )^3 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^3}-\frac{48 e^7 \left (a e^2+c d^2\right )}{d+e x}-192 c d e^7 \log \left (a+c x^2\right )+384 c d e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

((-48*e^7*(c*d^2 + a*e^2))/(d + e*x) + (3*c*(c*d^2 + a*e^2)*(5*c^3*d^6*x + 23*a*
c^2*d^4*e^2*x + 47*a^2*c*d^2*e^4*x + a^3*e^5*(48*d - 19*e*x)))/(a^3*(a + c*x^2))
+ (2*c*(c*d^2 + a*e^2)^2*(5*c^2*d^4*x + 18*a*c*d^2*e^2*x + a^2*e^3*(24*d - 11*e
*x)))/(a^2*(a + c*x^2)^2) + (8*c*(c*d^2 + a*e^2)^3*(c*d^2*x + a*e*(2*d - e*x)))/
(a*(a + c*x^2)^3) + (3*Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^
4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 384*c
*d*e^7*Log[d + e*x] - 192*c*d*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^5)

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Maple [B]  time = 0.033, size = 1130, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+a)^4,x)

[Out]

35/4*c^2/(a*e^2+c*d^2)^5/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^2*e^6-35/16*c/(a*
e^2+c*d^2)^5*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*e^8+5/16*c^5/(a*e^2+c*d^2)^5/
a^3/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^8-19/16*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^
3*a*x^5*e^8+7/4*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^5*d^2*e^6+7/4*c^4/(a*e^2+c*d^2
)^5/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^6*e^2+5/16*c^7/(a*e^2+c*d^2)^5/(c*
x^2+a)^3/a^3*x^5*d^8+3*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^4*d^3*e^5-17/6*c^2/(a*e
^2+c*d^2)^5/(c*x^2+a)^3*a^2*x^3*e^8+10*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^3*d^4*e
^4+5/6*c^6/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a^2*x^3*d^8+c^4/(a*e^2+c*d^2)^5/(c*x^2+a)
^3*x^2*d^5*e^3+13/4*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x*d^6*e^2+11/16*c^5/(a*e^2+c
*d^2)^5/(c*x^2+a)^3/a*x*d^8-29/16*c/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^3*x*e^8+13/3*c
/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^3*d*e^7+6*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^2*d^3
*e^5+2*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a*d^5*e^3+35/8*c^5/(a*e^2+c*d^2)^5/(c*x^2
+a)^3/a*x^5*d^4*e^4+7/4*c^6/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a^2*x^5*d^6*e^2+3*c^3/(a
*e^2+c*d^2)^5/(c*x^2+a)^3*x^4*a*d*e^7+10/3*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a*x^3
*d^2*e^6+14/3*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a*x^3*d^6*e^2+7*c^2/(a*e^2+c*d^2)^
5/(c*x^2+a)^3*x^2*a^2*d*e^7+8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^2*a*d^3*e^5+5/4*
c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^2*x*d^2*e^6+45/8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)
^3*a*x*d^4*e^4+35/8*c^3/(a*e^2+c*d^2)^5/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^
4*e^4+1/3*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*d^7*e-4*c/(a*e^2+c*d^2)^5*d*e^7*ln(a^3
*(c*x^2+a))-e^7/(a*e^2+c*d^2)^4/(e*x+d)+8*c*d*e^7*ln(e*x+d)/(a*e^2+c*d^2)^5

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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 + a)^4*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 39.9, 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 + a)^4*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/96*(32*a^3*c^4*d^8*e + 192*a^4*c^3*d^6*e^3 + 576*a^5*c^2*d^4*e^5 + 320*a^6*c*
d^2*e^7 - 96*a^7*e^9 + 6*(5*c^7*d^8*e + 28*a*c^6*d^6*e^3 + 70*a^2*c^5*d^4*e^5 +
12*a^3*c^4*d^2*e^7 - 35*a^4*c^3*e^9)*x^6 + 6*(5*c^7*d^9 + 28*a*c^6*d^7*e^2 + 70*
a^2*c^5*d^5*e^4 + 76*a^3*c^4*d^3*e^6 + 29*a^4*c^3*d*e^8)*x^5 + 16*(5*a*c^6*d^8*e
+ 28*a^2*c^5*d^6*e^3 + 78*a^3*c^4*d^4*e^5 + 20*a^4*c^3*d^2*e^7 - 35*a^5*c^2*e^9
)*x^4 + 16*(5*a*c^6*d^9 + 28*a^2*c^5*d^7*e^2 + 66*a^3*c^4*d^5*e^4 + 68*a^4*c^3*d
^3*e^6 + 25*a^5*c^2*d*e^8)*x^3 + 6*(11*a^2*c^5*d^8*e + 68*a^3*c^4*d^6*e^3 + 218*
a^4*c^3*d^4*e^5 + 84*a^5*c^2*d^2*e^7 - 77*a^6*c*e^9)*x^2 - 3*(5*a^3*c^4*d^9 + 28
*a^4*c^3*d^7*e^2 + 70*a^5*c^2*d^5*e^4 + 140*a^6*c*d^3*e^6 - 35*a^7*d*e^8 + (5*c^
7*d^8*e + 28*a*c^6*d^6*e^3 + 70*a^2*c^5*d^4*e^5 + 140*a^3*c^4*d^2*e^7 - 35*a^4*c
^3*e^9)*x^7 + (5*c^7*d^9 + 28*a*c^6*d^7*e^2 + 70*a^2*c^5*d^5*e^4 + 140*a^3*c^4*d
^3*e^6 - 35*a^4*c^3*d*e^8)*x^6 + 3*(5*a*c^6*d^8*e + 28*a^2*c^5*d^6*e^3 + 70*a^3*
c^4*d^4*e^5 + 140*a^4*c^3*d^2*e^7 - 35*a^5*c^2*e^9)*x^5 + 3*(5*a*c^6*d^9 + 28*a^
2*c^5*d^7*e^2 + 70*a^3*c^4*d^5*e^4 + 140*a^4*c^3*d^3*e^6 - 35*a^5*c^2*d*e^8)*x^4
+ 3*(5*a^2*c^5*d^8*e + 28*a^3*c^4*d^6*e^3 + 70*a^4*c^3*d^4*e^5 + 140*a^5*c^2*d^
2*e^7 - 35*a^6*c*e^9)*x^3 + 3*(5*a^2*c^5*d^9 + 28*a^3*c^4*d^7*e^2 + 70*a^4*c^3*d
^5*e^4 + 140*a^5*c^2*d^3*e^6 - 35*a^6*c*d*e^8)*x^2 + (5*a^3*c^4*d^8*e + 28*a^4*c
^3*d^6*e^3 + 70*a^5*c^2*d^4*e^5 + 140*a^6*c*d^2*e^7 - 35*a^7*e^9)*x)*sqrt(-c/a)*
log((c*x^2 - 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 2*(33*a^2*c^5*d^9 + 172*a^3*c^
4*d^7*e^2 + 366*a^4*c^3*d^5*e^4 + 348*a^5*c^2*d^3*e^6 + 121*a^6*c*d*e^8)*x - 384
*(a^3*c^4*d*e^8*x^7 + a^3*c^4*d^2*e^7*x^6 + 3*a^4*c^3*d*e^8*x^5 + 3*a^4*c^3*d^2*
e^7*x^4 + 3*a^5*c^2*d*e^8*x^3 + 3*a^5*c^2*d^2*e^7*x^2 + a^6*c*d*e^8*x + a^6*c*d^
2*e^7)*log(c*x^2 + a) + 768*(a^3*c^4*d*e^8*x^7 + a^3*c^4*d^2*e^7*x^6 + 3*a^4*c^3
*d*e^8*x^5 + 3*a^4*c^3*d^2*e^7*x^4 + 3*a^5*c^2*d*e^8*x^3 + 3*a^5*c^2*d^2*e^7*x^2
+ a^6*c*d*e^8*x + a^6*c*d^2*e^7)*log(e*x + d))/(a^6*c^5*d^11 + 5*a^7*c^4*d^9*e^
2 + 10*a^8*c^3*d^7*e^4 + 10*a^9*c^2*d^5*e^6 + 5*a^10*c*d^3*e^8 + a^11*d*e^10 + (
a^3*c^8*d^10*e + 5*a^4*c^7*d^8*e^3 + 10*a^5*c^6*d^6*e^5 + 10*a^6*c^5*d^4*e^7 + 5
*a^7*c^4*d^2*e^9 + a^8*c^3*e^11)*x^7 + (a^3*c^8*d^11 + 5*a^4*c^7*d^9*e^2 + 10*a^
5*c^6*d^7*e^4 + 10*a^6*c^5*d^5*e^6 + 5*a^7*c^4*d^3*e^8 + a^8*c^3*d*e^10)*x^6 + 3
*(a^4*c^7*d^10*e + 5*a^5*c^6*d^8*e^3 + 10*a^6*c^5*d^6*e^5 + 10*a^7*c^4*d^4*e^7 +
5*a^8*c^3*d^2*e^9 + a^9*c^2*e^11)*x^5 + 3*(a^4*c^7*d^11 + 5*a^5*c^6*d^9*e^2 + 1
0*a^6*c^5*d^7*e^4 + 10*a^7*c^4*d^5*e^6 + 5*a^8*c^3*d^3*e^8 + a^9*c^2*d*e^10)*x^4
+ 3*(a^5*c^6*d^10*e + 5*a^6*c^5*d^8*e^3 + 10*a^7*c^4*d^6*e^5 + 10*a^8*c^3*d^4*e
^7 + 5*a^9*c^2*d^2*e^9 + a^10*c*e^11)*x^3 + 3*(a^5*c^6*d^11 + 5*a^6*c^5*d^9*e^2
+ 10*a^7*c^4*d^7*e^4 + 10*a^8*c^3*d^5*e^6 + 5*a^9*c^2*d^3*e^8 + a^10*c*d*e^10)*x
^2 + (a^6*c^5*d^10*e + 5*a^7*c^4*d^8*e^3 + 10*a^8*c^3*d^6*e^5 + 10*a^9*c^2*d^4*e
^7 + 5*a^10*c*d^2*e^9 + a^11*e^11)*x), 1/48*(16*a^3*c^4*d^8*e + 96*a^4*c^3*d^6*e
^3 + 288*a^5*c^2*d^4*e^5 + 160*a^6*c*d^2*e^7 - 48*a^7*e^9 + 3*(5*c^7*d^8*e + 28*
a*c^6*d^6*e^3 + 70*a^2*c^5*d^4*e^5 + 12*a^3*c^4*d^2*e^7 - 35*a^4*c^3*e^9)*x^6 +
3*(5*c^7*d^9 + 28*a*c^6*d^7*e^2 + 70*a^2*c^5*d^5*e^4 + 76*a^3*c^4*d^3*e^6 + 29*a
^4*c^3*d*e^8)*x^5 + 8*(5*a*c^6*d^8*e + 28*a^2*c^5*d^6*e^3 + 78*a^3*c^4*d^4*e^5 +
20*a^4*c^3*d^2*e^7 - 35*a^5*c^2*e^9)*x^4 + 8*(5*a*c^6*d^9 + 28*a^2*c^5*d^7*e^2
+ 66*a^3*c^4*d^5*e^4 + 68*a^4*c^3*d^3*e^6 + 25*a^5*c^2*d*e^8)*x^3 + 3*(11*a^2*c^
5*d^8*e + 68*a^3*c^4*d^6*e^3 + 218*a^4*c^3*d^4*e^5 + 84*a^5*c^2*d^2*e^7 - 77*a^6
*c*e^9)*x^2 + 3*(5*a^3*c^4*d^9 + 28*a^4*c^3*d^7*e^2 + 70*a^5*c^2*d^5*e^4 + 140*a
^6*c*d^3*e^6 - 35*a^7*d*e^8 + (5*c^7*d^8*e + 28*a*c^6*d^6*e^3 + 70*a^2*c^5*d^4*e
^5 + 140*a^3*c^4*d^2*e^7 - 35*a^4*c^3*e^9)*x^7 + (5*c^7*d^9 + 28*a*c^6*d^7*e^2 +
70*a^2*c^5*d^5*e^4 + 140*a^3*c^4*d^3*e^6 - 35*a^4*c^3*d*e^8)*x^6 + 3*(5*a*c^6*d
^8*e + 28*a^2*c^5*d^6*e^3 + 70*a^3*c^4*d^4*e^5 + 140*a^4*c^3*d^2*e^7 - 35*a^5*c^
2*e^9)*x^5 + 3*(5*a*c^6*d^9 + 28*a^2*c^5*d^7*e^2 + 70*a^3*c^4*d^5*e^4 + 140*a^4*
c^3*d^3*e^6 - 35*a^5*c^2*d*e^8)*x^4 + 3*(5*a^2*c^5*d^8*e + 28*a^3*c^4*d^6*e^3 +
70*a^4*c^3*d^4*e^5 + 140*a^5*c^2*d^2*e^7 - 35*a^6*c*e^9)*x^3 + 3*(5*a^2*c^5*d^9
+ 28*a^3*c^4*d^7*e^2 + 70*a^4*c^3*d^5*e^4 + 140*a^5*c^2*d^3*e^6 - 35*a^6*c*d*e^8
)*x^2 + (5*a^3*c^4*d^8*e + 28*a^4*c^3*d^6*e^3 + 70*a^5*c^2*d^4*e^5 + 140*a^6*c*d
^2*e^7 - 35*a^7*e^9)*x)*sqrt(c/a)*arctan(c*x/(a*sqrt(c/a))) + (33*a^2*c^5*d^9 +
172*a^3*c^4*d^7*e^2 + 366*a^4*c^3*d^5*e^4 + 348*a^5*c^2*d^3*e^6 + 121*a^6*c*d*e^
8)*x - 192*(a^3*c^4*d*e^8*x^7 + a^3*c^4*d^2*e^7*x^6 + 3*a^4*c^3*d*e^8*x^5 + 3*a^
4*c^3*d^2*e^7*x^4 + 3*a^5*c^2*d*e^8*x^3 + 3*a^5*c^2*d^2*e^7*x^2 + a^6*c*d*e^8*x
+ a^6*c*d^2*e^7)*log(c*x^2 + a) + 384*(a^3*c^4*d*e^8*x^7 + a^3*c^4*d^2*e^7*x^6 +
3*a^4*c^3*d*e^8*x^5 + 3*a^4*c^3*d^2*e^7*x^4 + 3*a^5*c^2*d*e^8*x^3 + 3*a^5*c^2*d
^2*e^7*x^2 + a^6*c*d*e^8*x + a^6*c*d^2*e^7)*log(e*x + d))/(a^6*c^5*d^11 + 5*a^7*
c^4*d^9*e^2 + 10*a^8*c^3*d^7*e^4 + 10*a^9*c^2*d^5*e^6 + 5*a^10*c*d^3*e^8 + a^11*
d*e^10 + (a^3*c^8*d^10*e + 5*a^4*c^7*d^8*e^3 + 10*a^5*c^6*d^6*e^5 + 10*a^6*c^5*d
^4*e^7 + 5*a^7*c^4*d^2*e^9 + a^8*c^3*e^11)*x^7 + (a^3*c^8*d^11 + 5*a^4*c^7*d^9*e
^2 + 10*a^5*c^6*d^7*e^4 + 10*a^6*c^5*d^5*e^6 + 5*a^7*c^4*d^3*e^8 + a^8*c^3*d*e^1
0)*x^6 + 3*(a^4*c^7*d^10*e + 5*a^5*c^6*d^8*e^3 + 10*a^6*c^5*d^6*e^5 + 10*a^7*c^4
*d^4*e^7 + 5*a^8*c^3*d^2*e^9 + a^9*c^2*e^11)*x^5 + 3*(a^4*c^7*d^11 + 5*a^5*c^6*d
^9*e^2 + 10*a^6*c^5*d^7*e^4 + 10*a^7*c^4*d^5*e^6 + 5*a^8*c^3*d^3*e^8 + a^9*c^2*d
*e^10)*x^4 + 3*(a^5*c^6*d^10*e + 5*a^6*c^5*d^8*e^3 + 10*a^7*c^4*d^6*e^5 + 10*a^8
*c^3*d^4*e^7 + 5*a^9*c^2*d^2*e^9 + a^10*c*e^11)*x^3 + 3*(a^5*c^6*d^11 + 5*a^6*c^
5*d^9*e^2 + 10*a^7*c^4*d^7*e^4 + 10*a^8*c^3*d^5*e^6 + 5*a^9*c^2*d^3*e^8 + a^10*c
*d*e^10)*x^2 + (a^6*c^5*d^10*e + 5*a^7*c^4*d^8*e^3 + 10*a^8*c^3*d^6*e^5 + 10*a^9
*c^2*d^4*e^7 + 5*a^10*c*d^2*e^9 + a^11*e^11)*x)]

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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)**2/(c*x**2+a)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221506, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done