Optimal. Leaf size=71 \[ -\frac{b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac{b (2 c d-3 b e)}{c^4 (b+c x)}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]
[Out]
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Rubi [A] time = 0.15547, antiderivative size = 71, 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 \[ -\frac{b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac{b (2 c d-3 b e)}{c^4 (b+c x)}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(d + e*x))/(b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} \left (b e - c d\right )}{2 c^{4} \left (b + c x\right )^{2}} - \frac{b \left (3 b e - 2 c d\right )}{c^{4} \left (b + c x\right )} + \frac{\int e\, dx}{c^{3}} - \frac{\left (3 b e - c d\right ) \log{\left (b + c x \right )}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(e*x+d)/(c*x**2+b*x)**3,x)
[Out]
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Mathematica [A] time = 0.0497519, size = 75, normalized size = 1.06 \[ \frac{2 b c d-3 b^2 e}{c^4 (b+c x)}+\frac{b^3 e-b^2 c d}{2 c^4 (b+c x)^2}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(d + e*x))/(b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.011, size = 94, normalized size = 1.3 \[{\frac{ex}{{c}^{3}}}-3\,{\frac{\ln \left ( cx+b \right ) be}{{c}^{4}}}+{\frac{\ln \left ( cx+b \right ) d}{{c}^{3}}}+{\frac{{b}^{3}e}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}-{\frac{{b}^{2}d}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}-3\,{\frac{{b}^{2}e}{{c}^{4} \left ( cx+b \right ) }}+2\,{\frac{bd}{{c}^{3} \left ( cx+b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(e*x+d)/(c*x^2+b*x)^3,x)
[Out]
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Maxima [A] time = 0.696346, size = 112, normalized size = 1.58 \[ \frac{3 \, b^{2} c d - 5 \, b^{3} e + 2 \,{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} + \frac{e x}{c^{3}} + \frac{{\left (c d - 3 \, b e\right )} \log \left (c x + b\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^5/(c*x^2 + b*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270165, size = 177, normalized size = 2.49 \[ \frac{2 \, c^{3} e x^{3} + 4 \, b c^{2} e x^{2} + 3 \, b^{2} c d - 5 \, b^{3} e + 4 \,{\left (b c^{2} d - b^{2} c e\right )} x + 2 \,{\left (b^{2} c d - 3 \, b^{3} e +{\left (c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \,{\left (b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^5/(c*x^2 + b*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.84901, size = 83, normalized size = 1.17 \[ - \frac{5 b^{3} e - 3 b^{2} c d + x \left (6 b^{2} c e - 4 b c^{2} d\right )}{2 b^{2} c^{4} + 4 b c^{5} x + 2 c^{6} x^{2}} + \frac{e x}{c^{3}} - \frac{\left (3 b e - c d\right ) \log{\left (b + c x \right )}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(e*x+d)/(c*x**2+b*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.270347, size = 100, normalized size = 1.41 \[ \frac{x e}{c^{3}} + \frac{{\left (c d - 3 \, b e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{c^{4}} + \frac{3 \, b^{2} c d - 5 \, b^{3} e + 2 \,{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \,{\left (c x + b\right )}^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^5/(c*x^2 + b*x)^3,x, algorithm="giac")
[Out]