Optimal. Leaf size=106 \[ \frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b^2}{(d+e x) (b d-a e)^3}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}+\frac {1}{3 (d+e x)^3 (b d-a e)} \]
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Rubi [A] time = 0.07, antiderivative size = 106, 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 {b^2}{(d+e x) (b d-a e)^3}+\frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}+\frac {1}{3 (d+e x)^3 (b d-a e)} \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)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^4} \, dx\\ &=\int \left (\frac {b^4}{(b d-a e)^4 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^4}-\frac {b e}{(b d-a e)^2 (d+e x)^3}-\frac {b^2 e}{(b d-a e)^3 (d+e x)^2}-\frac {b^3 e}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=\frac {1}{3 (b d-a e) (d+e x)^3}+\frac {b}{2 (b d-a e)^2 (d+e x)^2}+\frac {b^2}{(b d-a e)^3 (d+e x)}+\frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 106, normalized size = 1.00 \begin {gather*} \frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b^2}{(d+e x) (b d-a e)^3}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}-\frac {1}{3 (d+e x)^3 (a e-b d)} \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)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 425, normalized size = 4.01 \begin {gather*} \frac {11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \, {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (b^{4} d^{7} - 4 \, a b^{3} d^{6} e + 6 \, a^{2} b^{2} d^{5} e^{2} - 4 \, a^{3} b d^{4} e^{3} + a^{4} d^{3} e^{4} + {\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{3} + 3 \, {\left (b^{4} d^{5} e^{2} - 4 \, a b^{3} d^{4} e^{3} + 6 \, a^{2} b^{2} d^{3} e^{4} - 4 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{2} + 3 \, {\left (b^{4} d^{6} e - 4 \, a b^{3} d^{5} e^{2} + 6 \, a^{2} b^{2} d^{4} e^{3} - 4 \, a^{3} b d^{3} e^{4} + a^{4} d^{2} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 238, normalized size = 2.25 \begin {gather*} \frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} - \frac {b^{3} e \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac {11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \, {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 104, normalized size = 0.98 \begin {gather*} \frac {b^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {b^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {b^{2}}{\left (a e -b d \right )^{3} \left (e x +d \right )}+\frac {b}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}}-\frac {1}{3 \left (a e -b d \right ) \left (e x +d \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 362, normalized size = 3.42 \begin {gather*} \frac {b^{3} \log \left (b x + a\right )}{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}} - \frac {b^{3} \log \left (e x + d\right )}{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}} + \frac {6 \, b^{2} e^{2} x^{2} + 11 \, b^{2} d^{2} - 7 \, a b d e + 2 \, a^{2} e^{2} + 3 \, {\left (5 \, b^{2} d e - a b e^{2}\right )} x}{6 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 313, normalized size = 2.95 \begin {gather*} \frac {2\,b^3\,\mathrm {atanh}\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,e\,x\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {2\,a^2\,e^2-7\,a\,b\,d\,e+11\,b^2\,d^2}{6\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {b\,x\,\left (a\,e^2-5\,b\,d\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b^2\,e^2\,x^2}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.53, size = 570, normalized size = 5.38 \begin {gather*} - \frac {b^{3} \log {\left (x + \frac {- \frac {a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} - \frac {5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e + \frac {b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac {b^{3} \log {\left (x + \frac {\frac {a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} + \frac {5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e - \frac {b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 a^{2} e^{2} + 7 a b d e - 11 b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} - 15 b^{2} d e\right )}{6 a^{3} d^{3} e^{3} - 18 a^{2} b d^{4} e^{2} + 18 a b^{2} d^{5} e - 6 b^{3} d^{6} + x^{3} \left (6 a^{3} e^{6} - 18 a^{2} b d e^{5} + 18 a b^{2} d^{2} e^{4} - 6 b^{3} d^{3} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{5} - 54 a^{2} b d^{2} e^{4} + 54 a b^{2} d^{3} e^{3} - 18 b^{3} d^{4} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{4} - 54 a^{2} b d^{3} e^{3} + 54 a b^{2} d^{4} e^{2} - 18 b^{3} d^{5} e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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