Optimal. Leaf size=179 \[ \frac {3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac {d^6}{2 b^3 x^2}+\frac {3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac {e^6 x}{c^3} \]
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Rubi [A] time = 0.22, antiderivative size = 179, 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^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac {3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}-\frac {d^6}{2 b^3 x^2}+\frac {e^6 x}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {e^6}{c^3}+\frac {d^6}{b^3 x^3}+\frac {3 d^5 (-c d+2 b e)}{b^4 x^2}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 x}-\frac {(-c d+b e)^6}{b^3 c^3 (b+c x)^3}+\frac {3 (-c d+b e)^5 (c d+b e)}{b^4 c^3 (b+c x)^2}-\frac {3 (-c d+b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 c^3 (b+c x)}\right ) \, dx\\ &=-\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 179, normalized size = 1.00 \begin {gather*} \frac {3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac {d^6}{2 b^3 x^2}+\frac {3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac {e^6 x}{c^3} \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)^6}{\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 = 0.44, size = 579, normalized size = 3.23 \begin {gather*} \frac {2 \, b^{5} c^{3} e^{6} x^{5} + 4 \, b^{6} c^{2} e^{6} x^{4} - b^{4} c^{4} d^{6} + 2 \, {\left (6 \, b c^{7} d^{6} - 18 \, b^{2} c^{6} d^{5} e + 15 \, b^{3} c^{5} d^{4} e^{2} - 15 \, b^{5} c^{3} d^{2} e^{4} + 12 \, b^{6} c^{2} d e^{5} - 2 \, b^{7} c e^{6}\right )} x^{3} + {\left (18 \, b^{2} c^{6} d^{6} - 54 \, b^{3} c^{5} d^{5} e + 45 \, b^{4} c^{4} d^{4} e^{2} - 20 \, b^{5} c^{3} d^{3} e^{3} - 15 \, b^{6} c^{2} d^{2} e^{4} + 18 \, b^{7} c d e^{5} - 5 \, b^{8} e^{6}\right )} x^{2} + 4 \, {\left (b^{3} c^{5} d^{6} - 3 \, b^{4} c^{4} d^{5} e\right )} x - 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2} - 2 \, b^{5} c^{3} d e^{5} + b^{6} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2} - 2 \, b^{6} c^{2} d e^{5} + b^{7} c e^{6}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2} - 2 \, b^{7} c d e^{5} + b^{8} e^{6}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{6} x^{4} + 2 \, b^{6} c^{5} x^{3} + b^{7} c^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 316, normalized size = 1.77 \begin {gather*} \frac {x e^{6}}{c^{3}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 396, normalized size = 2.21 \begin {gather*} \frac {b^{3} e^{6}}{2 \left (c x +b \right )^{2} c^{4}}-\frac {3 b^{2} d \,e^{5}}{\left (c x +b \right )^{2} c^{3}}+\frac {15 b \,d^{2} e^{4}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {15 d^{4} e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {3 c \,d^{5} e}{\left (c x +b \right )^{2} b^{2}}+\frac {c^{2} d^{6}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {10 d^{3} e^{3}}{\left (c x +b \right )^{2} c}-\frac {3 b^{2} e^{6}}{\left (c x +b \right ) c^{4}}+\frac {12 b d \,e^{5}}{\left (c x +b \right ) c^{3}}-\frac {3 b \,e^{6} \ln \left (c x +b \right )}{c^{4}}+\frac {15 d^{4} e^{2}}{\left (c x +b \right ) b^{2}}-\frac {12 c \,d^{5} e}{\left (c x +b \right ) b^{3}}+\frac {15 d^{4} e^{2} \ln \relax (x )}{b^{3}}-\frac {15 d^{4} e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 c^{2} d^{6}}{\left (c x +b \right ) b^{4}}-\frac {18 c \,d^{5} e \ln \relax (x )}{b^{4}}+\frac {18 c \,d^{5} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 c^{2} d^{6} \ln \relax (x )}{b^{5}}-\frac {6 c^{2} d^{6} \ln \left (c x +b \right )}{b^{5}}-\frac {15 d^{2} e^{4}}{\left (c x +b \right ) c^{2}}+\frac {6 d \,e^{5} \ln \left (c x +b \right )}{c^{3}}+\frac {e^{6} x}{c^{3}}-\frac {6 d^{5} e}{b^{3} x}+\frac {3 c \,d^{6}}{b^{4} x}-\frac {d^{6}}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 341, normalized size = 1.91 \begin {gather*} \frac {e^{6} x}{c^{3}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \relax (x)}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 333, normalized size = 1.86 \begin {gather*} \frac {e^6\,x}{c^3}-\frac {\frac {3\,x^3\,\left (b^6\,e^6-4\,b^5\,c\,d\,e^5+5\,b^4\,c^2\,d^2\,e^4-5\,b^2\,c^4\,d^4\,e^2+6\,b\,c^5\,d^5\,e-2\,c^6\,d^6\right )}{b^4}+\frac {c^3\,d^6}{2\,b}+\frac {x^2\,\left (5\,b^6\,e^6-18\,b^5\,c\,d\,e^5+15\,b^4\,c^2\,d^2\,e^4+20\,b^3\,c^3\,d^3\,e^3-45\,b^2\,c^4\,d^4\,e^2+54\,b\,c^5\,d^5\,e-18\,c^6\,d^6\right )}{2\,b^3\,c}+\frac {2\,c^3\,d^5\,x\,\left (3\,b\,e-c\,d\right )}{b^2}}{b^2\,c^3\,x^2+2\,b\,c^4\,x^3+c^5\,x^4}-\frac {\ln \left (b+c\,x\right )\,\left (3\,b^6\,e^6-6\,b^5\,c\,d\,e^5+15\,b^2\,c^4\,d^4\,e^2-18\,b\,c^5\,d^5\,e+6\,c^6\,d^6\right )}{b^5\,c^4}+\frac {3\,d^4\,\ln \relax (x)\,\left (5\,b^2\,e^2-6\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.86, size = 597, normalized size = 3.34 \begin {gather*} \frac {- b^{3} c^{4} d^{6} + x^{3} \left (- 6 b^{6} c e^{6} + 24 b^{5} c^{2} d e^{5} - 30 b^{4} c^{3} d^{2} e^{4} + 30 b^{2} c^{5} d^{4} e^{2} - 36 b c^{6} d^{5} e + 12 c^{7} d^{6}\right ) + x^{2} \left (- 5 b^{7} e^{6} + 18 b^{6} c d e^{5} - 15 b^{5} c^{2} d^{2} e^{4} - 20 b^{4} c^{3} d^{3} e^{3} + 45 b^{3} c^{4} d^{4} e^{2} - 54 b^{2} c^{5} d^{5} e + 18 b c^{6} d^{6}\right ) + x \left (- 12 b^{3} c^{4} d^{5} e + 4 b^{2} c^{5} d^{6}\right )}{2 b^{6} c^{4} x^{2} + 4 b^{5} c^{5} x^{3} + 2 b^{4} c^{6} x^{4}} + \frac {e^{6} x}{c^{3}} + \frac {3 d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} - 3 b c^{3} d^{4} \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right )}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5}} - \frac {3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} + \frac {3 b \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right )}{c}}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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