Optimal. Leaf size=143 \[ \frac {6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac {6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}+\frac {3 b^2 e}{(a+b x) (b d-a e)^4}-\frac {b^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {3 b e^2}{(d+e x) (b d-a e)^4}+\frac {e^2}{2 (d+e x)^2 (b d-a e)^3} \]
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Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 44} \begin {gather*} \frac {6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac {6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}+\frac {3 b^2 e}{(a+b x) (b d-a e)^4}-\frac {b^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {3 b e^2}{(d+e x) (b d-a e)^4}+\frac {e^2}{2 (d+e x)^2 (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^3 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^3}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^2}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)}-\frac {e^3}{(b d-a e)^3 (d+e x)^3}-\frac {3 b e^3}{(b d-a e)^4 (d+e x)^2}-\frac {6 b^2 e^3}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{2 (b d-a e)^3 (a+b x)^2}+\frac {3 b^2 e}{(b d-a e)^4 (a+b x)}+\frac {e^2}{2 (b d-a e)^3 (d+e x)^2}+\frac {3 b e^2}{(b d-a e)^4 (d+e x)}+\frac {6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac {6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 128, normalized size = 0.90 \begin {gather*} \frac {\frac {6 b^2 e (b d-a e)}{a+b x}-\frac {b^2 (b d-a e)^2}{(a+b x)^2}+12 b^2 e^2 \log (a+b x)+\frac {6 b e^2 (b d-a e)}{d+e x}+\frac {e^2 (b d-a e)^2}{(d+e x)^2}-12 b^2 e^2 \log (d+e x)}{2 (b d-a e)^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 {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 760, normalized size = 5.31 \begin {gather*} -\frac {b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \, {\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} + {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} + {\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \, {\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 332, normalized size = 2.32 \begin {gather*} \frac {6 \, b^{3} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} - \frac {6 \, b^{2} e^{3} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} + \frac {12 \, b^{3} x^{3} e^{3} + 18 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e - b^{3} d^{3} + 18 \, a b^{2} x^{2} e^{3} + 28 \, a b^{2} d x e^{2} + 7 \, a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + 7 \, a^{2} b d e^{2} - a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 140, normalized size = 0.98 \begin {gather*} -\frac {6 b^{2} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {6 b^{2} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {3 b \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {b^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {e^{2}}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 594, normalized size = 4.15 \begin {gather*} \frac {6 \, b^{2} e^{2} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {6 \, b^{2} e^{2} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {12 \, b^{3} e^{3} x^{3} - b^{3} d^{3} + 7 \, a b^{2} d^{2} e + 7 \, a^{2} b d e^{2} - a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + 7 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 542, normalized size = 3.79 \begin {gather*} \frac {\frac {6\,b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}-\frac {a^3\,e^3-7\,a^2\,b\,d\,e^2-7\,a\,b^2\,d^2\,e+b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,b\,e\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,b\,e\,x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4}-\frac {12\,b^2\,e^2\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.42, size = 881, normalized size = 6.16 \begin {gather*} \frac {6 b^{2} e^{2} \log {\left (x + \frac {- \frac {6 a^{6} b^{2} e^{8}}{\left (a e - b d\right )^{5}} + \frac {36 a^{5} b^{3} d e^{7}}{\left (a e - b d\right )^{5}} - \frac {90 a^{4} b^{4} d^{2} e^{6}}{\left (a e - b d\right )^{5}} + \frac {120 a^{3} b^{5} d^{3} e^{5}}{\left (a e - b d\right )^{5}} - \frac {90 a^{2} b^{6} d^{4} e^{4}}{\left (a e - b d\right )^{5}} + \frac {36 a b^{7} d^{5} e^{3}}{\left (a e - b d\right )^{5}} + 6 a b^{2} e^{3} - \frac {6 b^{8} d^{6} e^{2}}{\left (a e - b d\right )^{5}} + 6 b^{3} d e^{2}}{12 b^{3} e^{3}} \right )}}{\left (a e - b d\right )^{5}} - \frac {6 b^{2} e^{2} \log {\left (x + \frac {\frac {6 a^{6} b^{2} e^{8}}{\left (a e - b d\right )^{5}} - \frac {36 a^{5} b^{3} d e^{7}}{\left (a e - b d\right )^{5}} + \frac {90 a^{4} b^{4} d^{2} e^{6}}{\left (a e - b d\right )^{5}} - \frac {120 a^{3} b^{5} d^{3} e^{5}}{\left (a e - b d\right )^{5}} + \frac {90 a^{2} b^{6} d^{4} e^{4}}{\left (a e - b d\right )^{5}} - \frac {36 a b^{7} d^{5} e^{3}}{\left (a e - b d\right )^{5}} + 6 a b^{2} e^{3} + \frac {6 b^{8} d^{6} e^{2}}{\left (a e - b d\right )^{5}} + 6 b^{3} d e^{2}}{12 b^{3} e^{3}} \right )}}{\left (a e - b d\right )^{5}} + \frac {- a^{3} e^{3} + 7 a^{2} b d e^{2} + 7 a b^{2} d^{2} e - b^{3} d^{3} + 12 b^{3} e^{3} x^{3} + x^{2} \left (18 a b^{2} e^{3} + 18 b^{3} d e^{2}\right ) + x \left (4 a^{2} b e^{3} + 28 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{2 a^{6} d^{2} e^{4} - 8 a^{5} b d^{3} e^{3} + 12 a^{4} b^{2} d^{4} e^{2} - 8 a^{3} b^{3} d^{5} e + 2 a^{2} b^{4} d^{6} + x^{4} \left (2 a^{4} b^{2} e^{6} - 8 a^{3} b^{3} d e^{5} + 12 a^{2} b^{4} d^{2} e^{4} - 8 a b^{5} d^{3} e^{3} + 2 b^{6} d^{4} e^{2}\right ) + x^{3} \left (4 a^{5} b e^{6} - 12 a^{4} b^{2} d e^{5} + 8 a^{3} b^{3} d^{2} e^{4} + 8 a^{2} b^{4} d^{3} e^{3} - 12 a b^{5} d^{4} e^{2} + 4 b^{6} d^{5} e\right ) + x^{2} \left (2 a^{6} e^{6} - 18 a^{4} b^{2} d^{2} e^{4} + 32 a^{3} b^{3} d^{3} e^{3} - 18 a^{2} b^{4} d^{4} e^{2} + 2 b^{6} d^{6}\right ) + x \left (4 a^{6} d e^{5} - 12 a^{5} b d^{2} e^{4} + 8 a^{4} b^{2} d^{3} e^{3} + 8 a^{3} b^{3} d^{4} e^{2} - 12 a^{2} b^{4} d^{5} e + 4 a b^{5} d^{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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