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

Optimal. Leaf size=300 \[ -\frac{a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac{3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac{3 \sqrt{c} \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac{6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]

[Out]

(3*e*(c*d^2 - a*e^2)*(c*d^2 + 5*a*e^2))/(8*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) + (a
*e + c*d*x)/(4*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(c*d^2 - 5*a*e^
2) - 3*c*d*(c*d^2 + 3*a*e^2)*x)/(8*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*(a + c*x^2))
+ (3*Sqrt[c]*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTan[(
Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) + (6*c*d*e^5*Log[d + e*x])/(c
*d^2 + a*e^2)^4 - (3*c*d*e^5*Log[a + c*x^2])/(c*d^2 + a*e^2)^4

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Rubi [A]  time = 0.843026, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac{3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac{3 \sqrt{c} \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac{6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*(c*d^2 - a*e^2)*(c*d^2 + 5*a*e^2))/(8*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) + (a
*e + c*d*x)/(4*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^2) - (a*e*(c*d^2 - 5*a*e^
2) - 3*c*d*(c*d^2 + 3*a*e^2)*x)/(8*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*(a + c*x^2))
+ (3*Sqrt[c]*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTan[(
Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c*d^2 + a*e^2)^4) + (6*c*d*e^5*Log[d + e*x])/(c
*d^2 + a*e^2)^4 - (3*c*d*e^5*Log[a + c*x^2])/(c*d^2 + a*e^2)^4

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.803203, size = 241, normalized size = 0.8 \[ \frac{\frac{c \left (a e^2+c d^2\right ) \left (a^2 e^3 (16 d-7 e x)+12 a c d^2 e^2 x+3 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{3 \sqrt{c} \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 c \left (a e^2+c d^2\right )^2 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^2}-\frac{8 e^5 \left (a e^2+c d^2\right )}{d+e x}-24 c d e^5 \log \left (a+c x^2\right )+48 c d e^5 \log (d+e x)}{8 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

((-8*e^5*(c*d^2 + a*e^2))/(d + e*x) + (c*(c*d^2 + a*e^2)*(3*c^2*d^4*x + 12*a*c*d
^2*e^2*x + a^2*e^3*(16*d - 7*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(
c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*x^2)^2) + (3*Sqrt[c]*(c^3*d^6 + 5*a*c^2*d^
4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 48*
c*d*e^5*Log[d + e*x] - 24*c*d*e^5*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)

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Maple [B]  time = 0.027, size = 684, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+a)^3,x)

[Out]

-e^5/(a*e^2+c*d^2)^3/(e*x+d)+6*c*d*e^5*ln(e*x+d)/(a*e^2+c*d^2)^4-7/8*c^2/(a*e^2+
c*d^2)^4/(c*x^2+a)^2*a*x^3*e^6+5/8*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^3*d^2*e^4+1
5/8*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x^3*d^4*e^2+3/8*c^5/(a*e^2+c*d^2)^4/(c*x^2
+a)^2/a^2*x^3*d^6+2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x^2*a*d*e^5+2*c^3/(a*e^2+c*d
^2)^4/(c*x^2+a)^2*x^2*d^3*e^3-9/8*c/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a^2*x*e^6+3/8*c^
2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*x*d^2*e^4+17/8*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*x
*d^4*e^2+5/8*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x*d^6+5/2*c/(a*e^2+c*d^2)^4/(c*x^
2+a)^2*a^2*d*e^5+3*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a*d^3*e^3+1/2*c^3/(a*e^2+c*d^
2)^4/(c*x^2+a)^2*d^5*e-3*c/(a*e^2+c*d^2)^4*d*e^5*ln(a^2*(c*x^2+a))-15/8*c/(a*e^2
+c*d^2)^4*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*e^6+45/8*c^2/(a*e^2+c*d^2)^4/(a*
c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^2*e^4+15/8*c^3/(a*e^2+c*d^2)^4/a/(a*c)^(1/2)*
arctan(c*x/(a*c)^(1/2))*d^4*e^2+3/8*c^4/(a*e^2+c*d^2)^4/a^2/(a*c)^(1/2)*arctan(c
*x/(a*c)^(1/2))*d^6

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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)^3*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.92466, 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)^3*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/16*(8*a^2*c^3*d^6*e + 48*a^3*c^2*d^4*e^3 + 24*a^4*c*d^2*e^5 - 16*a^5*e^7 + 6*
(c^5*d^6*e + 5*a*c^4*d^4*e^3 - a^2*c^3*d^2*e^5 - 5*a^3*c^2*e^7)*x^4 + 6*(c^5*d^7
+ 5*a*c^4*d^5*e^2 + 7*a^2*c^3*d^3*e^4 + 3*a^3*c^2*d*e^6)*x^3 + 2*(5*a*c^4*d^6*e
+ 33*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - 25*a^4*c*e^7)*x^2 + 3*(a^2*c^3*d^7 +
5*a^3*c^2*d^5*e^2 + 15*a^4*c*d^3*e^4 - 5*a^5*d*e^6 + (c^5*d^6*e + 5*a*c^4*d^4*e
^3 + 15*a^2*c^3*d^2*e^5 - 5*a^3*c^2*e^7)*x^5 + (c^5*d^7 + 5*a*c^4*d^5*e^2 + 15*a
^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^4 + 2*(a*c^4*d^6*e + 5*a^2*c^3*d^4*e^3 + 15*
a^3*c^2*d^2*e^5 - 5*a^4*c*e^7)*x^3 + 2*(a*c^4*d^7 + 5*a^2*c^3*d^5*e^2 + 15*a^3*c
^2*d^3*e^4 - 5*a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e + 5*a^3*c^2*d^4*e^3 + 15*a^4*c*
d^2*e^5 - 5*a^5*e^7)*x)*sqrt(-c/a)*log((c*x^2 + 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a
)) + 2*(5*a*c^4*d^7 + 21*a^2*c^3*d^5*e^2 + 27*a^3*c^2*d^3*e^4 + 11*a^4*c*d*e^6)*
x - 48*(a^2*c^3*d*e^6*x^5 + a^2*c^3*d^2*e^5*x^4 + 2*a^3*c^2*d*e^6*x^3 + 2*a^3*c^
2*d^2*e^5*x^2 + a^4*c*d*e^6*x + a^4*c*d^2*e^5)*log(c*x^2 + a) + 96*(a^2*c^3*d*e^
6*x^5 + a^2*c^3*d^2*e^5*x^4 + 2*a^3*c^2*d*e^6*x^3 + 2*a^3*c^2*d^2*e^5*x^2 + a^4*
c*d*e^6*x + a^4*c*d^2*e^5)*log(e*x + d))/(a^4*c^4*d^9 + 4*a^5*c^3*d^7*e^2 + 6*a^
6*c^2*d^5*e^4 + 4*a^7*c*d^3*e^6 + a^8*d*e^8 + (a^2*c^6*d^8*e + 4*a^3*c^5*d^6*e^3
+ 6*a^4*c^4*d^4*e^5 + 4*a^5*c^3*d^2*e^7 + a^6*c^2*e^9)*x^5 + (a^2*c^6*d^9 + 4*a
^3*c^5*d^7*e^2 + 6*a^4*c^4*d^5*e^4 + 4*a^5*c^3*d^3*e^6 + a^6*c^2*d*e^8)*x^4 + 2*
(a^3*c^5*d^8*e + 4*a^4*c^4*d^6*e^3 + 6*a^5*c^3*d^4*e^5 + 4*a^6*c^2*d^2*e^7 + a^7
*c*e^9)*x^3 + 2*(a^3*c^5*d^9 + 4*a^4*c^4*d^7*e^2 + 6*a^5*c^3*d^5*e^4 + 4*a^6*c^2
*d^3*e^6 + a^7*c*d*e^8)*x^2 + (a^4*c^4*d^8*e + 4*a^5*c^3*d^6*e^3 + 6*a^6*c^2*d^4
*e^5 + 4*a^7*c*d^2*e^7 + a^8*e^9)*x), 1/8*(4*a^2*c^3*d^6*e + 24*a^3*c^2*d^4*e^3
+ 12*a^4*c*d^2*e^5 - 8*a^5*e^7 + 3*(c^5*d^6*e + 5*a*c^4*d^4*e^3 - a^2*c^3*d^2*e^
5 - 5*a^3*c^2*e^7)*x^4 + 3*(c^5*d^7 + 5*a*c^4*d^5*e^2 + 7*a^2*c^3*d^3*e^4 + 3*a^
3*c^2*d*e^6)*x^3 + (5*a*c^4*d^6*e + 33*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - 25*
a^4*c*e^7)*x^2 + 3*(a^2*c^3*d^7 + 5*a^3*c^2*d^5*e^2 + 15*a^4*c*d^3*e^4 - 5*a^5*d
*e^6 + (c^5*d^6*e + 5*a*c^4*d^4*e^3 + 15*a^2*c^3*d^2*e^5 - 5*a^3*c^2*e^7)*x^5 +
(c^5*d^7 + 5*a*c^4*d^5*e^2 + 15*a^2*c^3*d^3*e^4 - 5*a^3*c^2*d*e^6)*x^4 + 2*(a*c^
4*d^6*e + 5*a^2*c^3*d^4*e^3 + 15*a^3*c^2*d^2*e^5 - 5*a^4*c*e^7)*x^3 + 2*(a*c^4*d
^7 + 5*a^2*c^3*d^5*e^2 + 15*a^3*c^2*d^3*e^4 - 5*a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*
e + 5*a^3*c^2*d^4*e^3 + 15*a^4*c*d^2*e^5 - 5*a^5*e^7)*x)*sqrt(c/a)*arctan(c*x/(a
*sqrt(c/a))) + (5*a*c^4*d^7 + 21*a^2*c^3*d^5*e^2 + 27*a^3*c^2*d^3*e^4 + 11*a^4*c
*d*e^6)*x - 24*(a^2*c^3*d*e^6*x^5 + a^2*c^3*d^2*e^5*x^4 + 2*a^3*c^2*d*e^6*x^3 +
2*a^3*c^2*d^2*e^5*x^2 + a^4*c*d*e^6*x + a^4*c*d^2*e^5)*log(c*x^2 + a) + 48*(a^2*
c^3*d*e^6*x^5 + a^2*c^3*d^2*e^5*x^4 + 2*a^3*c^2*d*e^6*x^3 + 2*a^3*c^2*d^2*e^5*x^
2 + a^4*c*d*e^6*x + a^4*c*d^2*e^5)*log(e*x + d))/(a^4*c^4*d^9 + 4*a^5*c^3*d^7*e^
2 + 6*a^6*c^2*d^5*e^4 + 4*a^7*c*d^3*e^6 + a^8*d*e^8 + (a^2*c^6*d^8*e + 4*a^3*c^5
*d^6*e^3 + 6*a^4*c^4*d^4*e^5 + 4*a^5*c^3*d^2*e^7 + a^6*c^2*e^9)*x^5 + (a^2*c^6*d
^9 + 4*a^3*c^5*d^7*e^2 + 6*a^4*c^4*d^5*e^4 + 4*a^5*c^3*d^3*e^6 + a^6*c^2*d*e^8)*
x^4 + 2*(a^3*c^5*d^8*e + 4*a^4*c^4*d^6*e^3 + 6*a^5*c^3*d^4*e^5 + 4*a^6*c^2*d^2*e
^7 + a^7*c*e^9)*x^3 + 2*(a^3*c^5*d^9 + 4*a^4*c^4*d^7*e^2 + 6*a^5*c^3*d^5*e^4 + 4
*a^6*c^2*d^3*e^6 + a^7*c*d*e^8)*x^2 + (a^4*c^4*d^8*e + 4*a^5*c^3*d^6*e^3 + 6*a^6
*c^2*d^4*e^5 + 4*a^7*c*d^2*e^7 + a^8*e^9)*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218441, size = 768, normalized size = 2.56 \[ -\frac{3 \, c d e^{5}{\rm ln}\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{3 \,{\left (c^{4} d^{6} e^{2} + 5 \, a c^{3} d^{4} e^{4} + 15 \, a^{2} c^{2} d^{2} e^{6} - 5 \, a^{3} c e^{8}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{8 \,{\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt{a c}} - \frac{e^{11}}{{\left (c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}\right )}{\left (x e + d\right )}} + \frac{3 \, c^{5} d^{5} e + 14 \, a c^{4} d^{3} e^{3} - 29 \, a^{2} c^{3} d e^{5} - \frac{{\left (9 \, c^{5} d^{6} e^{2} + 41 \, a c^{4} d^{4} e^{4} - 121 \, a^{2} c^{3} d^{2} e^{6} + 7 \, a^{3} c^{2} e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac{{\left (9 \, c^{5} d^{7} e^{3} + 45 \, a c^{4} d^{5} e^{5} - 145 \, a^{2} c^{3} d^{3} e^{7} - 21 \, a^{3} c^{2} d e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{3 \,{\left (c^{5} d^{8} e^{4} + 6 \, a c^{4} d^{6} e^{6} - 20 \, a^{2} c^{3} d^{4} e^{8} - 22 \, a^{3} c^{2} d^{2} e^{10} + 3 \, a^{4} c e^{12}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \,{\left (c d^{2} + a e^{2}\right )}^{4} a^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*c*d*e^5*ln(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^4*
d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + 3/8*(c^
4*d^6*e^2 + 5*a*c^3*d^4*e^4 + 15*a^2*c^2*d^2*e^6 - 5*a^3*c*e^8)*arctan((c*d - c*
d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-1)/sqrt(a*c))*e^(-2)/((a^2*c^4*d^8 + 4*a^3*
c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(a*c)) - e^11/(
(c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*(x*e + d)) + 1/8*(
3*c^5*d^5*e + 14*a*c^4*d^3*e^3 - 29*a^2*c^3*d*e^5 - (9*c^5*d^6*e^2 + 41*a*c^4*d^
4*e^4 - 121*a^2*c^3*d^2*e^6 + 7*a^3*c^2*e^8)*e^(-1)/(x*e + d) + (9*c^5*d^7*e^3 +
45*a*c^4*d^5*e^5 - 145*a^2*c^3*d^3*e^7 - 21*a^3*c^2*d*e^9)*e^(-2)/(x*e + d)^2 -
3*(c^5*d^8*e^4 + 6*a*c^4*d^6*e^6 - 20*a^2*c^3*d^4*e^8 - 22*a^3*c^2*d^2*e^10 + 3
*a^4*c*e^12)*e^(-3)/(x*e + d)^3)/((c*d^2 + a*e^2)^4*a^2*(c - 2*c*d/(x*e + d) + c
*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)^2)