Optimal. Leaf size=110 \[ \frac {3 c \log (x) (2 c d-b e)}{b^5}-\frac {3 c (2 c d-b e) \log (b+c x)}{b^5}+\frac {3 c d-b e}{b^4 x}+\frac {c (3 c d-2 b e)}{b^4 (b+c x)}+\frac {c (c d-b e)}{2 b^3 (b+c x)^2}-\frac {d}{2 b^3 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \begin {gather*} \frac {3 c d-b e}{b^4 x}+\frac {c (3 c d-2 b e)}{b^4 (b+c x)}+\frac {c (c d-b e)}{2 b^3 (b+c x)^2}+\frac {3 c \log (x) (2 c d-b e)}{b^5}-\frac {3 c (2 c d-b e) \log (b+c x)}{b^5}-\frac {d}{2 b^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d}{b^3 x^3}+\frac {-3 c d+b e}{b^4 x^2}-\frac {3 c (-2 c d+b e)}{b^5 x}+\frac {c^2 (-c d+b e)}{b^3 (b+c x)^3}+\frac {c^2 (-3 c d+2 b e)}{b^4 (b+c x)^2}+\frac {3 c^2 (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {d}{2 b^3 x^2}+\frac {3 c d-b e}{b^4 x}+\frac {c (c d-b e)}{2 b^3 (b+c x)^2}+\frac {c (3 c d-2 b e)}{b^4 (b+c x)}+\frac {3 c (2 c d-b e) \log (x)}{b^5}-\frac {3 c (2 c d-b e) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 102, normalized size = 0.93 \begin {gather*} \frac {-\frac {b \left (b^3 (d+2 e x)+b^2 c x (9 e x-4 d)+6 b c^2 x^2 (e x-3 d)-12 c^3 d x^3\right )}{x^2 (b+c x)^2}+6 c \log (x) (2 c d-b e)+6 c (b e-2 c d) \log (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}{\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 = 234, normalized size = 2.13 \begin {gather*} -\frac {b^{4} d - 6 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} - 9 \, {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2} - 2 \, {\left (2 \, b^{3} c d - b^{4} e\right )} x + 6 \, {\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} + {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \, {\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} + {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 132, normalized size = 1.20 \begin {gather*} \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{3} d x^{3} - 6 \, b c^{2} x^{3} e + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c x^{2} e + 4 \, b^{2} c d x - 2 \, b^{3} x e - b^{3} d}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 138, normalized size = 1.25 \begin {gather*} -\frac {c e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d}{2 \left (c x +b \right )^{2} b^{3}}-\frac {2 c e}{\left (c x +b \right ) b^{3}}+\frac {3 c^{2} d}{\left (c x +b \right ) b^{4}}-\frac {3 c e \ln \relax (x )}{b^{4}}+\frac {3 c e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d \ln \left (c x +b \right )}{b^{5}}-\frac {e}{b^{3} x}+\frac {3 c d}{b^{4} x}-\frac {d}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 136, normalized size = 1.24 \begin {gather*} -\frac {b^{3} d - 6 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 9 \, {\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 2 \, {\left (2 \, b^{2} c d - b^{3} e\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\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 = 0.25, size = 132, normalized size = 1.20 \begin {gather*} -\frac {\frac {d}{2\,b}+\frac {x\,\left (b\,e-2\,c\,d\right )}{b^2}+\frac {9\,c\,x^2\,\left (b\,e-2\,c\,d\right )}{2\,b^3}+\frac {3\,c^2\,x^3\,\left (b\,e-2\,c\,d\right )}{b^4}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (b\,e-2\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (6\,c^2\,d-3\,b\,c\,e\right )}\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 = 0.73, size = 219, normalized size = 1.99 \begin {gather*} \frac {- b^{3} d + x^{3} \left (- 6 b c^{2} e + 12 c^{3} d\right ) + x^{2} \left (- 9 b^{2} c e + 18 b c^{2} d\right ) + x \left (- 2 b^{3} e + 4 b^{2} c d\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac {3 c \left (b e - 2 c d\right ) \log {\left (x + \frac {3 b^{2} c e - 6 b c^{2} d - 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} + \frac {3 c \left (b e - 2 c d\right ) \log {\left (x + \frac {3 b^{2} c e - 6 b c^{2} d + 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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