Optimal. Leaf size=94 \[ \frac {2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac {2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2} \]
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Rubi [A] time = 0.10, antiderivative size = 94, 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} \begin {gather*} -\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}+\frac {2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac {2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {e^4}{c^2}+\frac {d^4}{b^2 x^2}+\frac {2 d^3 (-c d+2 b e)}{b^3 x}+\frac {(-c d+b e)^4}{b^2 c^2 (b+c x)^2}-\frac {2 (-c d+b e)^3 (c d+b e)}{b^3 c^2 (b+c x)}\right ) \, dx\\ &=-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {2 d^3 (c d-2 b e) \log (x)}{b^3}+\frac {2 (c d-b e)^3 (c d+b e) \log (b+c x)}{b^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 95, normalized size = 1.01 \begin {gather*} \frac {2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}+\frac {2 d^3 \log (x) (2 b e-c d)}{b^3}-\frac {(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac {d^4}{b^2 x}+\frac {e^4 x}{c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 257, normalized size = 2.73 \begin {gather*} \frac {b^{3} c^{2} e^{4} x^{3} + b^{4} c e^{4} x^{2} - b^{2} c^{3} d^{4} - {\left (2 \, b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} x + 2 \, {\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e + 2 \, b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{2} + {\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e + 2 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e\right )} x^{2} + {\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e\right )} x\right )} \log \relax (x)}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 160, normalized size = 1.70 \begin {gather*} \frac {x e^{4}}{c^{2}} - \frac {2 \, {\left (c d^{4} - 2 \, b d^{3} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, {\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac {b c^{2} d^{4} + \frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 188, normalized size = 2.00 \begin {gather*} -\frac {b^{2} e^{4}}{\left (c x +b \right ) c^{3}}+\frac {4 b d \,e^{3}}{\left (c x +b \right ) c^{2}}-\frac {2 b \,e^{4} \ln \left (c x +b \right )}{c^{3}}+\frac {4 d^{3} e}{\left (c x +b \right ) b}-\frac {c \,d^{4}}{\left (c x +b \right ) b^{2}}+\frac {4 d^{3} e \ln \relax (x )}{b^{2}}-\frac {4 d^{3} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 c \,d^{4} \ln \relax (x )}{b^{3}}+\frac {2 c \,d^{4} \ln \left (c x +b \right )}{b^{3}}-\frac {6 d^{2} e^{2}}{\left (c x +b \right ) c}+\frac {4 d \,e^{3} \ln \left (c x +b \right )}{c^{2}}+\frac {e^{4} x}{c^{2}}-\frac {d^{4}}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 163, normalized size = 1.73 \begin {gather*} \frac {e^{4} x}{c^{2}} - \frac {b c^{3} d^{4} + {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} - \frac {2 \, {\left (c d^{4} - 2 \, b d^{3} e\right )} \log \relax (x)}{b^{3}} + \frac {2 \, {\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 166, normalized size = 1.77 \begin {gather*} \frac {e^4\,x}{c^2}-\frac {\frac {c^2\,d^4}{b}+\frac {x\,\left (b^4\,e^4-4\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{b^2\,c}}{c^3\,x^2+b\,c^2\,x}+\frac {2\,d^3\,\ln \relax (x)\,\left (2\,b\,e-c\,d\right )}{b^3}-\frac {\ln \left (b+c\,x\right )\,\left (2\,b^4\,e^4-4\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{b^3\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.76, size = 306, normalized size = 3.26 \begin {gather*} \frac {- b c^{3} d^{4} + x \left (- b^{4} e^{4} + 4 b^{3} c d e^{3} - 6 b^{2} c^{2} d^{2} e^{2} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac {e^{4} x}{c^{2}} + \frac {2 d^{3} \left (2 b e - c d\right ) \log {\left (x + \frac {4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} - 2 b c^{2} d^{3} \left (2 b e - c d\right )}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3}} - \frac {2 \left (b e - c d\right )^{3} \left (b e + c d\right ) \log {\left (x + \frac {4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} + \frac {2 b \left (b e - c d\right )^{3} \left (b e + c d\right )}{c}}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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