Optimal. Leaf size=144 \[ \frac{b^4 \log (a+b x)}{a^4 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{d^4 \log (c+d x)}{c^4 (b c-a d)}-\frac{1}{3 a c x^3} \]
[Out]
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Rubi [A] time = 0.272268, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{b^4 \log (a+b x)}{a^4 (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac{d^4 \log (c+d x)}{c^4 (b c-a d)}-\frac{1}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x)*(c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 55.6005, size = 128, normalized size = 0.89 \[ \frac{d^{4} \log{\left (c + d x \right )}}{c^{4} \left (a d - b c\right )} - \frac{1}{3 a c x^{3}} + \frac{a d + b c}{2 a^{2} c^{2} x^{2}} - \frac{a^{2} d^{2} + a b c d + b^{2} c^{2}}{a^{3} c^{3} x} - \frac{b^{4} \log{\left (a + b x \right )}}{a^{4} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \left (a^{2} d^{2} + b^{2} c^{2}\right ) \log{\left (x \right )}}{a^{4} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x+a)/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.0996833, size = 139, normalized size = 0.97 \[ \frac{6 x^3 \log (x) \left (b^4 c^4-a^4 d^4\right )+a \left (a^3 c d \left (-2 c^2+3 c d x-6 d^2 x^2\right )+6 a^3 d^4 x^3 \log (c+d x)+2 a^2 b c^4-3 a b^2 c^4 x+6 b^3 c^4 x^2\right )-6 b^4 c^4 x^3 \log (a+b x)}{6 a^4 c^4 x^3 (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x)*(c + d*x)),x]
[Out]
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Maple [A] time = 0.016, size = 179, normalized size = 1.2 \[{\frac{{d}^{4}\ln \left ( dx+c \right ) }{{c}^{4} \left ( ad-bc \right ) }}-{\frac{1}{3\,ac{x}^{3}}}+{\frac{d}{2\,{x}^{2}a{c}^{2}}}+{\frac{b}{2\,{a}^{2}{x}^{2}c}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}}-{\frac{\ln \left ( x \right ){d}^{3}}{a{c}^{4}}}-{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{3}}}-{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}{c}^{2}}}-{\frac{\ln \left ( x \right ){b}^{3}}{{a}^{4}c}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ){a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x+a)/(d*x+c),x)
[Out]
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Maxima [A] time = 1.35474, size = 211, normalized size = 1.47 \[ \frac{b^{4} \log \left (b x + a\right )}{a^{4} b c - a^{5} d} - \frac{d^{4} \log \left (d x + c\right )}{b c^{5} - a c^{4} d} - \frac{{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x\right )}{a^{4} c^{4}} - \frac{2 \, a^{2} c^{2} + 6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{2} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x}{6 \, a^{3} c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 3.85133, size = 201, normalized size = 1.4 \[ \frac{6 \, b^{4} c^{4} x^{3} \log \left (b x + a\right ) - 6 \, a^{4} d^{4} x^{3} \log \left (d x + c\right ) - 2 \, a^{3} b c^{4} + 2 \, a^{4} c^{3} d - 6 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x^{3} \log \left (x\right ) - 6 \,{\left (a b^{3} c^{4} - a^{4} c d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} c^{4} - a^{4} c^{2} d^{2}\right )} x}{6 \,{\left (a^{4} b c^{5} - a^{5} c^{4} d\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)*x^4),x, algorithm="fricas")
[Out]
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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/x**4/(b*x+a)/(d*x+c),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*(d*x + c)*x^4),x, algorithm="giac")
[Out]