Optimal. Leaf size=109 \[ \frac{e^2}{(d+e x) (b d-a e)^3}+\frac{3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac{3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac{2 b e}{(a+b x) (b d-a e)^3}-\frac{b}{2 (a+b x)^2 (b d-a e)^2} \]
[Out]
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Rubi [A] time = 0.162602, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{e^2}{(d+e x) (b d-a e)^3}+\frac{3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac{3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac{2 b e}{(a+b x) (b d-a e)^3}-\frac{b}{2 (a+b x)^2 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((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 = 67.1048, size = 97, normalized size = 0.89 \[ \frac{3 b e^{2} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{4}} - \frac{3 b e^{2} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{4}} - \frac{2 b e}{\left (a + b x\right ) \left (a e - b d\right )^{3}} - \frac{b}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} - \frac{e^{2}}{\left (d + e x\right ) \left (a e - b d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.132073, size = 98, normalized size = 0.9 \[ \frac{\frac{2 e^2 (b d-a e)}{d+e x}+\frac{4 b e (b d-a e)}{a+b x}-\frac{b (b d-a e)^2}{(a+b x)^2}+6 b e^2 \log (a+b x)-6 b e^2 \log (d+e x)}{2 (b d-a e)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 109, normalized size = 1. \[ -{\frac{b}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{b{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{b{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.730641, size = 521, normalized size = 4.78 \[ \frac{3 \, b e^{2} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{3 \, b e^{2} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{6 \, b^{2} e^{2} x^{2} - b^{2} d^{2} + 5 \, a b d e + 2 \, a^{2} e^{2} + 3 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{2 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((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 = 0.289245, size = 667, normalized size = 6.12 \[ -\frac{b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} +{\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} +{\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} +{\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((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 = 7.73281, size = 632, normalized size = 5.8 \[ - \frac{3 b e^{2} \log{\left (x + \frac{- \frac{3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} + \frac{15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} - \frac{30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} + \frac{30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} - \frac{15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} + \frac{3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac{3 b e^{2} \log{\left (x + \frac{\frac{3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} - \frac{15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} + \frac{30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} - \frac{30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} + \frac{15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} - \frac{3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} - \frac{2 a^{2} e^{2} + 5 a b d e - b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (9 a b e^{2} + 3 b^{2} d e\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(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 = 0.289433, size = 286, normalized size = 2.62 \[ \frac{3 \, b e^{3}{\rm ln}\left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac{e^{5}}{{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )}{\left (x e + d\right )}} + \frac{5 \, b^{3} e^{2} - \frac{6 \,{\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \,{\left (b d - a e\right )}^{4}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^2),x, algorithm="giac")
[Out]