Optimal. Leaf size=203 \[ \frac{(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{d^6 (3 c d-7 b e)}{b^4 x}+\frac{(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}-\frac{d^7}{2 b^3 x^2}+\frac{3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{2 c^3} \]
[Out]
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Rubi [A] time = 0.584017, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{d^6 (3 c d-7 b e)}{b^4 x}+\frac{(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}-\frac{d^7}{2 b^3 x^2}+\frac{3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac{3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^7/(b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - e^{6} \left (3 b e - 7 c d\right ) \int \frac{1}{c^{4}}\, dx + \frac{e^{7} \int x\, dx}{c^{3}} - \frac{d^{7}}{2 b^{3} x^{2}} - \frac{\left (b e - c d\right )^{7}}{2 b^{3} c^{5} \left (b + c x\right )^{2}} - \frac{d^{6} \left (7 b e - 3 c d\right )}{b^{4} x} + \frac{\left (b e - c d\right )^{6} \left (4 b e + 3 c d\right )}{b^{4} c^{5} \left (b + c x\right )} + \frac{3 d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x \right )}}{b^{5}} + \frac{3 \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**7/(c*x**2+b*x)**3,x)
[Out]
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Mathematica [A] time = 0.196687, size = 202, normalized size = 1. \[ \frac{1}{2} \left (\frac{2 (c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{2 d^6 (3 c d-7 b e)}{b^4 x}+\frac{(c d-b e)^7}{b^3 c^5 (b+c x)^2}-\frac{d^7}{b^3 x^2}+\frac{6 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}+\frac{6 (b e-c d)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac{2 e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{c^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^7/(b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.026, size = 481, normalized size = 2.4 \[ -{\frac{{d}^{7}}{2\,{b}^{3}{x}^{2}}}+{\frac{{e}^{7}{x}^{2}}{2\,{c}^{3}}}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{7}}{{c}^{5}}}+21\,{\frac{\ln \left ( cx+b \right ){d}^{2}{e}^{5}}{{c}^{3}}}-21\,{\frac{\ln \left ( cx+b \right ){d}^{5}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ){d}^{7}}{{b}^{5}}}+4\,{\frac{{b}^{3}{e}^{7}}{{c}^{5} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{7}}{{b}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{4}{e}^{7}}{2\,{c}^{5} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{7}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{21\,{d}^{5}{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{35\,{d}^{4}{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}-35\,{\frac{{d}^{3}{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+21\,{\frac{{d}^{5}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-3\,{\frac{b{e}^{7}x}{{c}^{4}}}+7\,{\frac{d{e}^{6}x}{{c}^{3}}}-7\,{\frac{{d}^{6}e}{{b}^{3}x}}+3\,{\frac{{d}^{7}c}{{b}^{4}x}}+21\,{\frac{{d}^{5}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{7}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}+{\frac{7\,d{b}^{3}{e}^{6}}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}-{\frac{21\,{b}^{2}{d}^{2}{e}^{5}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{35\,b{d}^{3}{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-21\,{\frac{{d}^{6}\ln \left ( x \right ) ce}{{b}^{4}}}-21\,{\frac{b\ln \left ( cx+b \right ) d{e}^{6}}{{c}^{4}}}+21\,{\frac{c\ln \left ( cx+b \right ){d}^{6}e}{{b}^{4}}}-21\,{\frac{{b}^{2}d{e}^{6}}{{c}^{4} \left ( cx+b \right ) }}+42\,{\frac{b{d}^{2}{e}^{5}}{{c}^{3} \left ( cx+b \right ) }}-{\frac{7\,c{d}^{6}e}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}-14\,{\frac{c{d}^{6}e}{{b}^{3} \left ( cx+b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^7/(c*x^2+b*x)^3,x)
[Out]
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Maxima [A] time = 0.711948, size = 551, normalized size = 2.71 \[ -\frac{b^{3} c^{5} d^{7} - 2 \,{\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} -{\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \,{\left (b^{4} c^{7} x^{4} + 2 \, b^{5} c^{6} x^{3} + b^{6} c^{5} x^{2}\right )}} + \frac{c e^{7} x^{2} + 2 \,{\left (7 \, c d e^{6} - 3 \, b e^{7}\right )} x}{2 \, c^{4}} + \frac{3 \,{\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left (c x + b\right )}{b^{5} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*x^2 + b*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266601, size = 937, normalized size = 4.62 \[ \frac{b^{5} c^{4} e^{7} x^{6} - b^{4} c^{5} d^{7} + 2 \,{\left (7 \, b^{5} c^{4} d e^{6} - 2 \, b^{6} c^{3} e^{7}\right )} x^{5} +{\left (28 \, b^{6} c^{3} d e^{6} - 11 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \,{\left (6 \, b c^{8} d^{7} - 21 \, b^{2} c^{7} d^{6} e + 21 \, b^{3} c^{6} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{3} e^{4} + 42 \, b^{6} c^{3} d^{2} e^{5} - 14 \, b^{7} c^{2} d e^{6} + b^{8} c e^{7}\right )} x^{3} +{\left (18 \, b^{2} c^{7} d^{7} - 63 \, b^{3} c^{6} d^{6} e + 63 \, b^{4} c^{5} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{4} e^{3} - 35 \, b^{6} c^{3} d^{3} e^{4} + 63 \, b^{7} c^{2} d^{2} e^{5} - 35 \, b^{8} c d e^{6} + 7 \, b^{9} e^{7}\right )} x^{2} + 2 \,{\left (2 \, b^{3} c^{6} d^{7} - 7 \, b^{4} c^{5} d^{6} e\right )} x - 6 \,{\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2} - 7 \, b^{5} c^{4} d^{2} e^{5} + 7 \, b^{6} c^{3} d e^{6} - 2 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \,{\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2} - 7 \, b^{6} c^{3} d^{2} e^{5} + 7 \, b^{7} c^{2} d e^{6} - 2 \, b^{8} c e^{7}\right )} x^{3} +{\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2} - 7 \, b^{7} c^{2} d^{2} e^{5} + 7 \, b^{8} c d e^{6} - 2 \, b^{9} e^{7}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{7} x^{4} + 2 \, b^{6} c^{6} x^{3} + b^{7} c^{5} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*x^2 + b*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 98.0873, size = 685, normalized size = 3.37 \[ \frac{- b^{3} c^{5} d^{7} + x^{3} \left (8 b^{7} c e^{7} - 42 b^{6} c^{2} d e^{6} + 84 b^{5} c^{3} d^{2} e^{5} - 70 b^{4} c^{4} d^{3} e^{4} + 42 b^{2} c^{6} d^{5} e^{2} - 42 b c^{7} d^{6} e + 12 c^{8} d^{7}\right ) + x^{2} \left (7 b^{8} e^{7} - 35 b^{7} c d e^{6} + 63 b^{6} c^{2} d^{2} e^{5} - 35 b^{5} c^{3} d^{3} e^{4} - 35 b^{4} c^{4} d^{4} e^{3} + 63 b^{3} c^{5} d^{5} e^{2} - 63 b^{2} c^{6} d^{6} e + 18 b c^{7} d^{7}\right ) + x \left (- 14 b^{3} c^{5} d^{6} e + 4 b^{2} c^{6} d^{7}\right )}{2 b^{6} c^{5} x^{2} + 4 b^{5} c^{6} x^{3} + 2 b^{4} c^{7} x^{4}} + \frac{e^{7} x^{2}}{2 c^{3}} - \frac{x \left (3 b e^{7} - 7 c d e^{6}\right )}{c^{4}} + \frac{3 d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + 3 b c^{4} d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right )}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5}} + \frac{3 \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right ) \log{\left (x + \frac{- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + \frac{3 b \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right )}{c}}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**7/(c*x**2+b*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.211559, size = 517, normalized size = 2.55 \[ \frac{3 \,{\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{c^{3} x^{2} e^{7} + 14 \, c^{3} d x e^{6} - 6 \, b c^{2} x e^{7}}{2 \, c^{6}} - \frac{3 \,{\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{5}} - \frac{b^{3} c^{5} d^{7} - 2 \,{\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} -{\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*x^2 + b*x)^3,x, algorithm="giac")
[Out]