Optimal. Leaf size=77 \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]
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Rubi [A] time = 0.130673, antiderivative size = 77, 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 \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x)*(c + d*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} \log{\left (a + b x \right )}}{b^{3} \left (a d - b c\right )} - \frac{c^{3} \log{\left (c + d x \right )}}{d^{3} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \int \frac{1}{b^{2}}\, dx}{d^{2}} + \frac{\int x\, dx}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x+a)/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.0588202, size = 74, normalized size = 0.96 \[ \frac{-2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (-2 a d-2 b c+b d x)+2 b^3 c^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x)*(c + d*x)),x]
[Out]
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Maple [A] time = 0.009, size = 80, normalized size = 1. \[{\frac{{x}^{2}}{2\,bd}}-{\frac{ax}{{b}^{2}d}}-{\frac{cx}{b{d}^{2}}}-{\frac{{c}^{3}\ln \left ( dx+c \right ) }{{d}^{3} \left ( ad-bc \right ) }}+{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{3} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x+a)/(d*x+c),x)
[Out]
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Maxima [A] time = 1.36947, size = 104, normalized size = 1.35 \[ -\frac{a^{3} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} + \frac{c^{3} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac{b d x^{2} - 2 \,{\left (b c + a d\right )} x}{2 \, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)*(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221891, size = 128, normalized size = 1.66 \[ -\frac{2 \, a^{3} d^{3} \log \left (b x + a\right ) - 2 \, b^{3} c^{3} \log \left (d x + c\right ) -{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x}{2 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.82358, size = 219, normalized size = 2.84 \[ \frac{a^{3} \log{\left (x + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )} - \frac{2 a^{4} c d^{3}}{a d - b c} + \frac{a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{b^{3} \left (a d - b c\right )} - \frac{c^{3} \log{\left (x + \frac{a^{3} c d^{2} - \frac{a^{2} b^{2} c^{3} d}{a d - b c} + \frac{2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac{b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} - \frac{x \left (a d + b c\right )}{b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(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(x^3/((b*x + a)*(d*x + c)),x, algorithm="giac")
[Out]