Optimal. Leaf size=107 \[ -\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}-\frac{d^2}{(a+b x) (b c-a d)^3}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)} \]
[Out]
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Rubi [A] time = 0.172236, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}-\frac{d^2}{(a+b x) (b c-a d)^3}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 39.8619, size = 88, normalized size = 0.82 \[ - \frac{d^{3} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{4}} + \frac{d^{3} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{4}} + \frac{d^{2}}{\left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{d}{2 \left (a + b x\right )^{2} \left (a d - b c\right )^{2}} + \frac{1}{3 \left (a + b x\right )^{3} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [A] time = 0.07923, size = 107, normalized size = 1. \[ -\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}-\frac{d^2}{(a+b x) (b c-a d)^3}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}+\frac{1}{3 (a+b x)^3 (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
[Out]
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Maple [A] time = 0.016, size = 103, normalized size = 1. \[{\frac{1}{ \left ( 3\,ad-3\,bc \right ) \left ( bx+a \right ) ^{3}}}+{\frac{d}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{d}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}+{\frac{{d}^{3}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^3/(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.774352, size = 487, normalized size = 4.55 \[ -\frac{d^{3} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{d^{3} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{6 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211995, size = 574, normalized size = 5.36 \[ -\frac{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} +{\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \,{\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.7631, size = 570, normalized size = 5.33 \[ \frac{d^{3} \log{\left (x + \frac{- \frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} + \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} - \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} + \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} - \frac{d^{3} \log{\left (x + \frac{\frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} - \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} + \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} - \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} + \frac{11 a^{2} d^{2} - 7 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (15 a b d^{2} - 3 b^{2} c d\right )}{6 a^{6} d^{3} - 18 a^{5} b c d^{2} + 18 a^{4} b^{2} c^{2} d - 6 a^{3} b^{3} c^{3} + x^{3} \left (6 a^{3} b^{3} d^{3} - 18 a^{2} b^{4} c d^{2} + 18 a b^{5} c^{2} d - 6 b^{6} c^{3}\right ) + x^{2} \left (18 a^{4} b^{2} d^{3} - 54 a^{3} b^{3} c d^{2} + 54 a^{2} b^{4} c^{2} d - 18 a b^{5} c^{3}\right ) + x \left (18 a^{5} b d^{3} - 54 a^{4} b^{2} c d^{2} + 54 a^{3} b^{3} c^{2} d - 18 a^{2} b^{4} c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.214074, size = 328, normalized size = 3.07 \[ -\frac{b d^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac{d^{4}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x}{6 \,{\left (b c - a d\right )}^{4}{\left (b x + a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3),x, algorithm="giac")
[Out]