Optimal. Leaf size=86 \[ -\frac {3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac {3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac {(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac {e^3 \log (a+b x)}{b^4} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac {3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac {(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac {e^3 \log (a+b x)}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^3}{(a+b x)^4} \, dx\\ &=\int \left (\frac {(b d-a e)^3}{b^3 (a+b x)^4}+\frac {3 e (b d-a e)^2}{b^3 (a+b x)^3}+\frac {3 e^2 (b d-a e)}{b^3 (a+b x)^2}+\frac {e^3}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac {(b d-a e)^3}{3 b^4 (a+b x)^3}-\frac {3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac {3 e^2 (b d-a e)}{b^4 (a+b x)}+\frac {e^3 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.93 \begin {gather*} \frac {6 e^3 \log (a+b x)-\frac {(b d-a e) \left (11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a+b x)^3}}{6 b^4} \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 (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 = 0.39, size = 176, normalized size = 2.05 \begin {gather*} -\frac {2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 113, normalized size = 1.31 \begin {gather*} \frac {e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {18 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 9 \, {\left (b^{2} d^{2} e + 2 \, a b d e^{2} - 3 \, a^{2} e^{3}\right )} x + \frac {2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 166, normalized size = 1.93 \begin {gather*} \frac {a^{3} e^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {a^{2} d \,e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {a \,d^{2} e}{\left (b x +a \right )^{3} b^{2}}-\frac {d^{3}}{3 \left (b x +a \right )^{3} b}-\frac {3 a^{2} e^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {3 a d \,e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {3 d^{2} e}{2 \left (b x +a \right )^{2} b^{2}}+\frac {3 a \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {3 d \,e^{2}}{\left (b x +a \right ) b^{3}}+\frac {e^{3} \ln \left (b x +a \right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 142, normalized size = 1.65 \begin {gather*} -\frac {2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {e^{3} \log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 138, normalized size = 1.60 \begin {gather*} \frac {e^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {\frac {-11\,a^3\,e^3+6\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+2\,b^3\,d^3}{6\,b^4}+\frac {3\,x\,\left (-3\,a^2\,e^3+2\,a\,b\,d\,e^2+b^2\,d^2\,e\right )}{2\,b^3}-\frac {3\,e^2\,x^2\,\left (a\,e-b\,d\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.14, size = 148, normalized size = 1.72 \begin {gather*} \frac {11 a^{3} e^{3} - 6 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 2 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (27 a^{2} b e^{3} - 18 a b^{2} d e^{2} - 9 b^{3} d^{2} e\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {e^{3} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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