Optimal. Leaf size=103 \[ \frac {4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac {6 e^2 (b d-a e)^2}{b^5 (a+b x)}-\frac {2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac {(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac {e^4 x}{b^4} \]
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Rubi [A] time = 0.09, antiderivative size = 103, 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 {6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac {4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac {2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac {(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac {e^4 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)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^4}{(a+b x)^4} \, dx\\ &=\int \left (\frac {e^4}{b^4}+\frac {(b d-a e)^4}{b^4 (a+b x)^4}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^3}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)^2}+\frac {4 e^3 (b d-a e)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {e^4 x}{b^4}-\frac {(b d-a e)^4}{3 b^5 (a+b x)^3}-\frac {2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac {6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac {4 e^3 (b d-a e) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 166, normalized size = 1.61 \begin {gather*} \frac {-13 a^4 e^4+a^3 b e^3 (22 d-27 e x)-3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a b^3 e \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 (a+b x)^3 (a e-b d) \log (a+b x)-\left (b^4 \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )}{3 b^5 (a+b x)^3} \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)^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 = 2.83 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} + 9 \, a b^{3} e^{4} x^{3} - b^{4} d^{4} - 2 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} + 22 \, a^{3} b d e^{3} - 13 \, a^{4} e^{4} - 9 \, {\left (2 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} - 3 \, {\left (2 \, b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 18 \, a^{2} b^{2} d e^{3} + 9 \, a^{3} b e^{4}\right )} x + 12 \, {\left (a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 167, normalized size = 1.62 \begin {gather*} \frac {x e^{4}}{b^{4}} + \frac {4 \, {\left (b d e^{3} - a e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 255, normalized size = 2.48 \begin {gather*} -\frac {a^{4} e^{4}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {4 a^{3} d \,e^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {2 a^{2} d^{2} e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {4 a \,d^{3} e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {d^{4}}{3 \left (b x +a \right )^{3} b}+\frac {2 a^{3} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {6 a^{2} d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}+\frac {6 a \,d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {2 d^{3} e}{\left (b x +a \right )^{2} b^{2}}-\frac {6 a^{2} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {12 a d \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {4 a \,e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {6 d^{2} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {4 d \,e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {e^{4} x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 201, normalized size = 1.95 \begin {gather*} \frac {e^{4} x}{b^{4}} - \frac {b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {4 \, {\left (b d e^{3} - a 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 = 0.66, size = 204, normalized size = 1.98 \begin {gather*} \frac {e^4\,x}{b^4}-\frac {\ln \left (a+b\,x\right )\,\left (4\,a\,e^4-4\,b\,d\,e^3\right )}{b^5}-\frac {\frac {13\,a^4\,e^4-22\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4}{3\,b}+x\,\left (10\,a^3\,e^4-18\,a^2\,b\,d\,e^3+6\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )+x^2\,\left (6\,a^2\,b\,e^4-12\,a\,b^2\,d\,e^3+6\,b^3\,d^2\,e^2\right )}{a^3\,b^4+3\,a^2\,b^5\,x+3\,a\,b^6\,x^2+b^7\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.94, size = 209, normalized size = 2.03 \begin {gather*} \frac {- 13 a^{4} e^{4} + 22 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 2 a b^{3} d^{3} e - b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 30 a^{3} b e^{4} + 54 a^{2} b^{2} d e^{3} - 18 a b^{3} d^{2} e^{2} - 6 b^{4} d^{3} e\right )}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac {e^{4} x}{b^{4}} - \frac {4 e^{3} \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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