Optimal. Leaf size=132 \[ -\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2} \]
[Out]
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Rubi [A] time = 0.257488, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 99.6351, size = 117, normalized size = 0.89 \[ \frac{4 b e^{3} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} - \frac{4 b e^{3} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} - \frac{3 b e^{2}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b e}{\left (a + b x\right )^{2} \left (a e - b d\right )^{3}} - \frac{b}{3 \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} - \frac{e^{3}}{\left (d + e x\right ) \left (a e - b d\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.225222, size = 121, normalized size = 0.92 \[ \frac{\frac{3 e^3 (a e-b d)}{d+e x}-\frac{9 b e^2 (b d-a e)}{a+b x}+\frac{3 b e (b d-a e)^2}{(a+b x)^2}-\frac{b (b d-a e)^3}{(a+b x)^3}-12 b e^3 \log (a+b x)+12 b e^3 \log (d+e x)}{3 (b d-a e)^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
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Maple [A] time = 0.021, size = 132, normalized size = 1. \[ -{\frac{b}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}+4\,{\frac{b{e}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{b{e}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.715306, size = 807, normalized size = 6.11 \[ -\frac{4 \, b e^{3} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{4 \, b e^{3} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{12 \, b^{3} e^{3} x^{3} + b^{3} d^{3} - 5 \, a b^{2} d^{2} e + 13 \, a^{2} b d e^{2} + 3 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (b^{3} d^{2} e - 8 \, a b^{2} d e^{2} - 11 \, a^{2} b e^{3}\right )} x}{3 \,{\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} +{\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} +{\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \,{\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} +{\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217493, size = 1014, normalized size = 7.68 \[ -\frac{b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} +{\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} +{\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} +{\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.7651, size = 881, normalized size = 6.67 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.21712, size = 377, normalized size = 2.86 \[ -\frac{4 \, b e^{4}{\rm ln}\left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{e^{7}}{{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}{\left (x e + d\right )}} - \frac{13 \, b^{4} e^{3} - \frac{30 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{3 \,{\left (b d - a e\right )}^{5}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^2),x, algorithm="giac")
[Out]