Optimal. Leaf size=111 \[ -\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5} \]
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Rubi [A] time = 0.10, antiderivative size = 111, 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 {4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5} \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)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^4}{(a+b x)^5} \, dx\\ &=\int \left (\frac {(b d-a e)^4}{b^4 (a+b x)^5}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^4}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)^3}+\frac {4 e^3 (b d-a e)}{b^4 (a+b x)^2}+\frac {e^4}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}+\frac {e^4 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 120, normalized size = 1.08 \begin {gather*} \frac {e^4 \log (a+b x)}{b^5}-\frac {(b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^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 {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 267, normalized size = 2.41 \begin {gather*} -\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 174, normalized size = 1.57 \begin {gather*} \frac {e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {48 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \, {\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 260, normalized size = 2.34 \begin {gather*} -\frac {a^{4} e^{4}}{4 \left (b x +a \right )^{4} b^{5}}+\frac {a^{3} d \,e^{3}}{\left (b x +a \right )^{4} b^{4}}-\frac {3 a^{2} d^{2} e^{2}}{2 \left (b x +a \right )^{4} b^{3}}+\frac {a \,d^{3} e}{\left (b x +a \right )^{4} b^{2}}-\frac {d^{4}}{4 \left (b x +a \right )^{4} b}+\frac {4 a^{3} e^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {4 a^{2} d \,e^{3}}{\left (b x +a \right )^{3} b^{4}}+\frac {4 a \,d^{2} e^{2}}{\left (b x +a \right )^{3} b^{3}}-\frac {4 d^{3} e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {3 a^{2} e^{4}}{\left (b x +a \right )^{2} b^{5}}+\frac {6 a d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {3 d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {4 a \,e^{4}}{\left (b x +a \right ) b^{5}}-\frac {4 d \,e^{3}}{\left (b x +a \right ) b^{4}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 219, normalized size = 1.97 \begin {gather*} -\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac {e^{4} \log \left (b x + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.09, size = 213, normalized size = 1.92 \begin {gather*} \frac {e^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {\frac {-25\,a^4\,e^4+12\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+3\,b^4\,d^4}{12\,b^5}+\frac {3\,x^2\,\left (-3\,a^2\,e^4+2\,a\,b\,d\,e^3+b^2\,d^2\,e^2\right )}{b^3}+\frac {2\,x\,\left (-11\,a^3\,e^4+6\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )}{3\,b^4}-\frac {4\,e^3\,x^3\,\left (a\,e-b\,d\right )}{b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.34, size = 230, normalized size = 2.07 \begin {gather*} \frac {25 a^{4} e^{4} - 12 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - 3 b^{4} d^{4} + x^{3} \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \left (108 a^{2} b^{2} e^{4} - 72 a b^{3} d e^{3} - 36 b^{4} d^{2} e^{2}\right ) + x \left (88 a^{3} b e^{4} - 48 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} - 16 b^{4} d^{3} e\right )}{12 a^{4} b^{5} + 48 a^{3} b^{6} x + 72 a^{2} b^{7} x^{2} + 48 a b^{8} x^{3} + 12 b^{9} x^{4}} + \frac {e^{4} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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