Optimal. Leaf size=102 \[ \frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}+\frac {e^3 x (4 b d-3 a e)}{b^4}+\frac {e^4 x^2}{2 b^3} \]
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Rubi [A] time = 0.09, antiderivative size = 102, 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 {e^3 x (4 b d-3 a e)}{b^4}+\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}+\frac {e^4 x^2}{2 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)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^4}{(a+b x)^3} \, dx\\ &=\int \left (\frac {e^3 (4 b d-3 a e)}{b^4}+\frac {e^4 x}{b^3}+\frac {(b d-a e)^4}{b^4 (a+b x)^3}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^2}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {e^3 (4 b d-3 a e) x}{b^4}+\frac {e^4 x^2}{2 b^3}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}+\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 163, normalized size = 1.60 \begin {gather*} \frac {7 a^4 e^4+2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+12 e^2 (a+b x)^2 (b d-a e)^2 \log (a+b x)+b^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )}{2 b^5 (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)^4}{\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 = 0.41, size = 292, normalized size = 2.86 \begin {gather*} \frac {b^{4} e^{4} x^{4} - b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 20 \, a^{3} b d e^{3} + 7 \, a^{4} e^{4} + 4 \, {\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + {\left (16 \, a b^{3} d e^{3} - 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (4 \, b^{4} d^{3} e - 12 \, a b^{3} d^{2} e^{2} + 8 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 172, normalized size = 1.69 \begin {gather*} \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} e^{4} + 8 \, b^{3} d x e^{3} - 6 \, a b^{2} x e^{4}}{2 \, b^{6}} - \frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 245, normalized size = 2.40 \begin {gather*} -\frac {a^{4} e^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {2 a^{3} d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2} d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {2 a \,d^{3} e}{\left (b x +a \right )^{2} b^{2}}-\frac {d^{4}}{2 \left (b x +a \right )^{2} b}+\frac {e^{4} x^{2}}{2 b^{3}}+\frac {4 a^{3} e^{4}}{\left (b x +a \right ) b^{5}}-\frac {12 a^{2} d \,e^{3}}{\left (b x +a \right ) b^{4}}+\frac {6 a^{2} e^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {12 a \,d^{2} e^{2}}{\left (b x +a \right ) b^{3}}-\frac {12 a d \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 a \,e^{4} x}{b^{4}}-\frac {4 d^{3} e}{\left (b x +a \right ) b^{2}}+\frac {6 d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {4 d \,e^{3} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 190, normalized size = 1.86 \begin {gather*} -\frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b e^{4} x^{2} + 2 \, {\left (4 \, b d e^{3} - 3 \, a e^{4}\right )} x}{2 \, b^{4}} + \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \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 = 0.10, size = 197, normalized size = 1.93 \begin {gather*} \frac {e^4\,x^2}{2\,b^3}-x\,\left (\frac {3\,a\,e^4}{b^4}-\frac {4\,d\,e^3}{b^3}\right )-\frac {\frac {-7\,a^4\,e^4+20\,a^3\,b\,d\,e^3-18\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{2\,b}-x\,\left (4\,a^3\,e^4-12\,a^2\,b\,d\,e^3+12\,a\,b^2\,d^2\,e^2-4\,b^3\,d^3\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,a^2\,e^4-12\,a\,b\,d\,e^3+6\,b^2\,d^2\,e^2\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 185, normalized size = 1.81 \begin {gather*} x \left (- \frac {3 a e^{4}}{b^{4}} + \frac {4 d e^{3}}{b^{3}}\right ) + \frac {7 a^{4} e^{4} - 20 a^{3} b d e^{3} + 18 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - b^{4} d^{4} + x \left (8 a^{3} b e^{4} - 24 a^{2} b^{2} d e^{3} + 24 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {e^{4} x^{2}}{2 b^{3}} + \frac {6 e^{2} \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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