Optimal. Leaf size=78 \[ \frac {3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac {3 e (b d-a e)^2}{b^4 (a+b x)}-\frac {(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac {e^3 x}{b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 78, 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, 43} \begin {gather*} \frac {3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac {3 e (b d-a e)^2}{b^4 (a+b x)}-\frac {(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac {e^3 x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
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 {(d+e x)^3}{(a+b x)^3} \, dx\\ &=\int \left (\frac {e^3}{b^3}+\frac {(b d-a e)^3}{b^3 (a+b x)^3}+\frac {3 e (b d-a e)^2}{b^3 (a+b x)^2}+\frac {3 e^2 (b d-a e)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {e^3 x}{b^3}-\frac {(b d-a e)^3}{2 b^4 (a+b x)^2}-\frac {3 e (b d-a e)^2}{b^4 (a+b x)}+\frac {3 e^2 (b d-a e) \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 114, normalized size = 1.46 \begin {gather*} \frac {-5 a^3 e^3+a^2 b e^2 (9 d-4 e x)+a b^2 e \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 (a+b x)^2 (a e-b d) \log (a+b x)-\left (b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )}{2 b^4 (a+b x)^2} \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 = 0.40, size = 188, normalized size = 2.41 \begin {gather*} \frac {2 \, b^{3} e^{3} x^{3} + 4 \, a b^{2} e^{3} x^{2} - b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - 2 \, {\left (3 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 2 \, a^{2} b e^{3}\right )} x + 6 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 107, normalized size = 1.37 \begin {gather*} \frac {x e^{3}}{b^{3}} + \frac {3 \, {\left (b d e^{2} - a e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 160, normalized size = 2.05 \begin {gather*} \frac {a^{3} e^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2} d \,e^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {3 a \,d^{2} e}{2 \left (b x +a \right )^{2} b^{2}}-\frac {d^{3}}{2 \left (b x +a \right )^{2} b}-\frac {3 a^{2} e^{3}}{\left (b x +a \right ) b^{4}}+\frac {6 a d \,e^{2}}{\left (b x +a \right ) b^{3}}-\frac {3 a \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 d^{2} e}{\left (b x +a \right ) b^{2}}+\frac {3 d \,e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {e^{3} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 125, normalized size = 1.60 \begin {gather*} \frac {e^{3} x}{b^{3}} - \frac {b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {3 \, {\left (b d e^{2} - a e^{3}\right )} \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 = 0.10, size = 130, normalized size = 1.67 \begin {gather*} \frac {e^3\,x}{b^3}-\frac {\ln \left (a+b\,x\right )\,\left (3\,a\,e^3-3\,b\,d\,e^2\right )}{b^4}-\frac {\frac {5\,a^3\,e^3-9\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+b^3\,d^3}{2\,b}+x\,\left (3\,a^2\,e^3-6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 128, normalized size = 1.64 \begin {gather*} \frac {- 5 a^{3} e^{3} + 9 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - b^{3} d^{3} + x \left (- 6 a^{2} b e^{3} + 12 a b^{2} d e^{2} - 6 b^{3} d^{2} e\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {e^{3} x}{b^{3}} - \frac {3 e^{2} \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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