Optimal. Leaf size=45 \[ \frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}+\frac{e x}{c^2} \]
[Out]
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Rubi [A] time = 0.0992133, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}+\frac{e x}{c^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(d + e*x))/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{b \left (b e - c d\right )}{c^{3} \left (b + c x\right )} + \frac{\int e\, dx}{c^{2}} - \frac{\left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(e*x+d)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.0480368, size = 41, normalized size = 0.91 \[ \frac{\frac{b (c d-b e)}{b+c x}+(c d-2 b e) \log (b+c x)+c e x}{c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(d + e*x))/(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 61, normalized size = 1.4 \[{\frac{ex}{{c}^{2}}}-2\,{\frac{\ln \left ( cx+b \right ) be}{{c}^{3}}}+{\frac{\ln \left ( cx+b \right ) d}{{c}^{2}}}-{\frac{{b}^{2}e}{{c}^{3} \left ( cx+b \right ) }}+{\frac{bd}{{c}^{2} \left ( cx+b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(e*x+d)/(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.708424, size = 68, normalized size = 1.51 \[ \frac{b c d - b^{2} e}{c^{4} x + b c^{3}} + \frac{e x}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )} \log \left (c x + b\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^3/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271258, size = 93, normalized size = 2.07 \[ \frac{c^{2} e x^{2} + b c e x + b c d - b^{2} e +{\left (b c d - 2 \, b^{2} e +{\left (c^{2} d - 2 \, b c e\right )} x\right )} \log \left (c x + b\right )}{c^{4} x + b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^3/(c*x^2 + b*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.8173, size = 44, normalized size = 0.98 \[ - \frac{b^{2} e - b c d}{b c^{3} + c^{4} x} + \frac{e x}{c^{2}} - \frac{\left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(e*x+d)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.269485, size = 69, normalized size = 1.53 \[ \frac{x e}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{3}} + \frac{b c d - b^{2} e}{{\left (c x + b\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^3/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]