Optimal. Leaf size=137 \[ \frac {3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}+\frac {3 d^2 (c d-b e)}{b^4 x}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}-\frac {d^3}{2 b^3 x^2} \]
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Rubi [A] time = 0.15, antiderivative size = 137, 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 {3 d^2 (c d-b e)}{b^4 x}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac {3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}-\frac {d^3}{2 b^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d^3}{b^3 x^3}+\frac {3 d^2 (-c d+b e)}{b^4 x^2}+\frac {3 d (c d-b e) (2 c d-b e)}{b^5 x}+\frac {(-c d+b e)^3}{b^3 (b+c x)^3}-\frac {3 c d (-c d+b e)^2}{b^4 (b+c x)^2}+\frac {3 c d (c d-b e) (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {d^3}{2 b^3 x^2}+\frac {3 d^2 (c d-b e)}{b^4 x}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {3 d (c d-b e) (2 c d-b e) \log (x)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 138, normalized size = 1.01 \begin {gather*} -\frac {-6 d \log (x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (b+c x)+\frac {b^2 (b e-c d)^3}{c (b+c x)^2}+\frac {b^2 d^3}{x^2}+\frac {6 b d^2 (b e-c d)}{x}-\frac {6 b d (c d-b e)^2}{b+c x}}{2 b^5} \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)^3}{\left (b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 385, normalized size = 2.81 \begin {gather*} -\frac {b^{4} c d^{3} - 6 \, {\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} - {\left (18 \, b^{2} c^{3} d^{3} - 27 \, b^{3} c^{2} d^{2} e + 9 \, b^{4} c d e^{2} - b^{5} e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{2} d^{3} - 3 \, b^{4} c d^{2} e\right )} x + 6 \, {\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \, {\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 219, normalized size = 1.60 \begin {gather*} \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 9 \, b^{3} c d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - b^{3} c d^{3} - b^{4} x^{2} e^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 238, normalized size = 1.74 \begin {gather*} \frac {3 d \,e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {3 c \,d^{2} e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{3}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {e^{3}}{2 \left (c x +b \right )^{2} c}+\frac {3 d \,e^{2}}{\left (c x +b \right ) b^{2}}-\frac {6 c \,d^{2} e}{\left (c x +b \right ) b^{3}}+\frac {3 d \,e^{2} \ln \relax (x )}{b^{3}}-\frac {3 d \,e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{3}}{\left (c x +b \right ) b^{4}}-\frac {9 c \,d^{2} e \ln \relax (x )}{b^{4}}+\frac {9 c \,d^{2} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{3} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{3} \ln \left (c x +b \right )}{b^{5}}-\frac {3 d^{2} e}{b^{3} x}+\frac {3 c \,d^{3}}{b^{4} x}-\frac {d^{3}}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 217, normalized size = 1.58 \begin {gather*} -\frac {b^{3} c d^{3} - 6 \, {\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{2} c^{2} d e^{2}\right )} x^{3} - {\left (18 \, b c^{3} d^{3} - 27 \, b^{2} c^{2} d^{2} e + 9 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{2} d^{3} - 3 \, b^{3} c d^{2} e\right )} x}{2 \, {\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \relax (x)}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 211, normalized size = 1.54 \begin {gather*} -\frac {\frac {d^3}{2\,b}+\frac {d^2\,x\,\left (3\,b\,e-2\,c\,d\right )}{b^2}+\frac {x^2\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+27\,b\,c^2\,d^2\,e-18\,c^3\,d^3\right )}{2\,b^3\,c}-\frac {3\,c\,d\,x^3\,\left (b^2\,e^2-3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^4}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,d\,\mathrm {atanh}\left (\frac {3\,d\,\left (b\,e-c\,d\right )\,\left (b\,e-2\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (3\,b^2\,d\,e^2-9\,b\,c\,d^2\,e+6\,c^2\,d^3\right )}\right )\,\left (b\,e-c\,d\right )\,\left (b\,e-2\,c\,d\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.76, size = 371, normalized size = 2.71 \begin {gather*} \frac {- b^{3} c d^{3} + x^{3} \left (6 b^{2} c^{2} d e^{2} - 18 b c^{3} d^{2} e + 12 c^{4} d^{3}\right ) + x^{2} \left (- b^{4} e^{3} + 9 b^{3} c d e^{2} - 27 b^{2} c^{2} d^{2} e + 18 b c^{3} d^{3}\right ) + x \left (- 6 b^{3} c d^{2} e + 4 b^{2} c^{2} d^{3}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac {3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (x + \frac {3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} - 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} - \frac {3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (x + \frac {3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} + 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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