Optimal. Leaf size=118 \[ \frac {(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac {d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac {(c d-b e)^5}{b^2 c^4 (b+c x)}-\frac {d^5}{b^2 x}+\frac {e^4 x (5 c d-2 b e)}{c^3}+\frac {e^5 x^2}{2 c^2} \]
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Rubi [A] time = 0.14, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {(c d-b e)^5}{b^2 c^4 (b+c x)}+\frac {(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac {d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac {d^5}{b^2 x}+\frac {e^4 x (5 c d-2 b e)}{c^3}+\frac {e^5 x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {e^4 (5 c d-2 b e)}{c^3}+\frac {d^5}{b^2 x^2}+\frac {d^4 (-2 c d+5 b e)}{b^3 x}+\frac {e^5 x}{c^2}-\frac {(-c d+b e)^5}{b^2 c^3 (b+c x)^2}+\frac {(-c d+b e)^4 (2 c d+3 b e)}{b^3 c^3 (b+c x)}\right ) \, dx\\ &=-\frac {d^5}{b^2 x}+\frac {e^4 (5 c d-2 b e) x}{c^3}+\frac {e^5 x^2}{2 c^2}-\frac {(c d-b e)^5}{b^2 c^4 (b+c x)}-\frac {d^4 (2 c d-5 b e) \log (x)}{b^3}+\frac {(c d-b e)^4 (2 c d+3 b e) \log (b+c x)}{b^3 c^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 116, normalized size = 0.98 \[ \frac {(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}+\frac {d^4 \log (x) (5 b e-2 c d)}{b^3}+\frac {(b e-c d)^5}{b^2 c^4 (b+c x)}-\frac {d^5}{b^2 x}+\frac {e^4 x (5 c d-2 b e)}{c^3}+\frac {e^5 x^2}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 349, normalized size = 2.96 \[ \frac {b^{3} c^{3} e^{5} x^{4} - 2 \, b^{2} c^{4} d^{5} + {\left (10 \, b^{3} c^{3} d e^{4} - 3 \, b^{4} c^{2} e^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{4} c^{2} d e^{4} - 2 \, b^{5} c e^{5}\right )} x^{2} - 2 \, {\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} + 5 \, b^{5} c d e^{4} - b^{6} e^{5}\right )} x + 2 \, {\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e + 10 \, b^{3} c^{3} d^{2} e^{3} - 10 \, b^{4} c^{2} d e^{4} + 3 \, b^{5} c e^{5}\right )} x^{2} + {\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e + 10 \, b^{4} c^{2} d^{2} e^{3} - 10 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (2 \, c^{6} d^{5} - 5 \, b c^{5} d^{4} e\right )} x^{2} + {\left (2 \, b c^{5} d^{5} - 5 \, b^{2} c^{4} d^{4} e\right )} x\right )} \log \relax (x)}{2 \, {\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 209, normalized size = 1.77 \[ -\frac {{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4} - 4 \, b c x e^{5}}{2 \, c^{4}} + \frac {{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac {b c^{4} d^{5} + {\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 251, normalized size = 2.13 \[ \frac {e^{5} x^{2}}{2 c^{2}}+\frac {b^{3} e^{5}}{\left (c x +b \right ) c^{4}}-\frac {5 b^{2} d \,e^{4}}{\left (c x +b \right ) c^{3}}+\frac {3 b^{2} e^{5} \ln \left (c x +b \right )}{c^{4}}+\frac {10 b \,d^{2} e^{3}}{\left (c x +b \right ) c^{2}}-\frac {10 b d \,e^{4} \ln \left (c x +b \right )}{c^{3}}-\frac {2 b \,e^{5} x}{c^{3}}+\frac {5 d^{4} e}{\left (c x +b \right ) b}-\frac {c \,d^{5}}{\left (c x +b \right ) b^{2}}+\frac {5 d^{4} e \ln \relax (x )}{b^{2}}-\frac {5 d^{4} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 c \,d^{5} \ln \relax (x )}{b^{3}}+\frac {2 c \,d^{5} \ln \left (c x +b \right )}{b^{3}}-\frac {10 d^{3} e^{2}}{\left (c x +b \right ) c}+\frac {10 d^{2} e^{3} \ln \left (c x +b \right )}{c^{2}}+\frac {5 d \,e^{4} x}{c^{2}}-\frac {d^{5}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 216, normalized size = 1.83 \[ -\frac {b c^{4} d^{5} + {\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - 10 \, b^{3} c^{2} d^{2} e^{3} + 5 \, b^{4} c d e^{4} - b^{5} e^{5}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} - \frac {{\left (2 \, c d^{5} - 5 \, b d^{4} e\right )} \log \relax (x)}{b^{3}} + \frac {c e^{5} x^{2} + 2 \, {\left (5 \, c d e^{4} - 2 \, b e^{5}\right )} x}{2 \, c^{3}} + \frac {{\left (2 \, c^{5} d^{5} - 5 \, b c^{4} d^{4} e + 10 \, b^{3} c^{2} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 218, normalized size = 1.85 \[ \frac {e^5\,x^2}{2\,c^2}-\frac {\frac {c^3\,d^5}{b}-\frac {x\,\left (b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^3\,c^2\,d^2\,e^3-10\,b^2\,c^3\,d^3\,e^2+5\,b\,c^4\,d^4\,e-2\,c^5\,d^5\right )}{b^2\,c}}{c^4\,x^2+b\,c^3\,x}-x\,\left (\frac {2\,b\,e^5}{c^3}-\frac {5\,d\,e^4}{c^2}\right )+\frac {\ln \left (b+c\,x\right )\,\left (3\,b^5\,e^5-10\,b^4\,c\,d\,e^4+10\,b^3\,c^2\,d^2\,e^3-5\,b\,c^4\,d^4\,e+2\,c^5\,d^5\right )}{b^3\,c^4}+\frac {d^4\,\ln \relax (x)\,\left (5\,b\,e-2\,c\,d\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.96, size = 381, normalized size = 3.23 \[ x \left (- \frac {2 b e^{5}}{c^{3}} + \frac {5 d e^{4}}{c^{2}}\right ) + \frac {- b c^{4} d^{5} + x \left (b^{5} e^{5} - 5 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b^{2} c^{3} d^{3} e^{2} + 5 b c^{4} d^{4} e - 2 c^{5} d^{5}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} + \frac {e^{5} x^{2}}{2 c^{2}} + \frac {d^{4} \left (5 b e - 2 c d\right ) \log {\left (x + \frac {- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + b c^{3} d^{4} \left (5 b e - 2 c d\right )}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3}} + \frac {\left (b e - c d\right )^{4} \left (3 b e + 2 c d\right ) \log {\left (x + \frac {- 5 b^{2} c^{3} d^{4} e + 2 b c^{4} d^{5} + \frac {b \left (b e - c d\right )^{4} \left (3 b e + 2 c d\right )}{c}}{3 b^{5} e^{5} - 10 b^{4} c d e^{4} + 10 b^{3} c^{2} d^{2} e^{3} - 10 b c^{4} d^{4} e + 4 c^{5} d^{5}} \right )}}{b^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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