Optimal. Leaf size=104 \[ \frac {2 e^3 (a+b x)^2 (b d-a e)}{b^5}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {6 e^2 x (b d-a e)^2}{b^4} \]
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Rubi [A] time = 0.10, antiderivative size = 104, 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 {2 e^3 (a+b x)^2 (b d-a e)}{b^5}+\frac {6 e^2 x (b d-a e)^2}{b^4}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^4}{(a+b x)^2} \, dx\\ &=\int \left (\frac {6 e^2 (b d-a e)^2}{b^4}+\frac {(b d-a e)^4}{b^4 (a+b x)^2}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)}+\frac {4 e^3 (b d-a e) (a+b x)}{b^4}+\frac {e^4 (a+b x)^2}{b^4}\right ) \, dx\\ &=\frac {6 e^2 (b d-a e)^2 x}{b^4}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {2 e^3 (b d-a e) (a+b x)^2}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 166, normalized size = 1.60 \begin {gather*} \frac {-3 a^4 e^4+3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )-2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )-12 e (a+b x) (a e-b d)^3 \log (a+b x)+b^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )}{3 b^5 (a+b x)} \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}{a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 268, normalized size = 2.58 \begin {gather*} \frac {b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \, {\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\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\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a 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 = 178, normalized size = 1.71 \begin {gather*} \frac {4 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{4} x^{3} e^{4} + 6 \, b^{4} d x^{2} e^{3} + 18 \, b^{4} d^{2} x e^{2} - 3 \, a b^{3} x^{2} e^{4} - 24 \, a b^{3} d x e^{3} + 9 \, a^{2} b^{2} x e^{4}}{3 \, b^{6}} - \frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{{\left (b x + a\right )} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 230, normalized size = 2.21 \begin {gather*} \frac {e^{4} x^{3}}{3 b^{2}}-\frac {a \,e^{4} x^{2}}{b^{3}}+\frac {2 d \,e^{3} x^{2}}{b^{2}}-\frac {a^{4} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {4 a^{3} d \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {4 a^{3} e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {6 a^{2} d^{2} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {12 a^{2} d \,e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 a^{2} e^{4} x}{b^{4}}+\frac {4 a \,d^{3} e}{\left (b x +a \right ) b^{2}}-\frac {12 a \,d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {8 a d \,e^{3} x}{b^{3}}-\frac {d^{4}}{\left (b x +a \right ) b}+\frac {4 d^{3} e \ln \left (b x +a \right )}{b^{2}}+\frac {6 d^{2} e^{2} x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 184, normalized size = 1.77 \begin {gather*} -\frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{b^{6} x + a b^{5}} + \frac {b^{2} e^{4} x^{3} + 3 \, {\left (2 \, b^{2} d e^{3} - a b e^{4}\right )} x^{2} + 3 \, {\left (6 \, b^{2} d^{2} e^{2} - 8 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} x}{3 \, b^{4}} + \frac {4 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} 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 = 0.54, size = 203, normalized size = 1.95 \begin {gather*} x\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^4}{b^3}-\frac {4\,d\,e^3}{b^2}\right )}{b}-\frac {a^2\,e^4}{b^4}+\frac {6\,d^2\,e^2}{b^2}\right )-x^2\,\left (\frac {a\,e^4}{b^3}-\frac {2\,d\,e^3}{b^2}\right )+\frac {e^4\,x^3}{3\,b^2}-\frac {\ln \left (a+b\,x\right )\,\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 )}{b^5}-\frac {a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}{b\,\left (x\,b^5+a\,b^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 155, normalized size = 1.49 \begin {gather*} x^{2} \left (- \frac {a e^{4}}{b^{3}} + \frac {2 d e^{3}}{b^{2}}\right ) + x \left (\frac {3 a^{2} e^{4}}{b^{4}} - \frac {8 a d e^{3}}{b^{3}} + \frac {6 d^{2} e^{2}}{b^{2}}\right ) + \frac {- a^{4} e^{4} + 4 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 4 a b^{3} d^{3} e - b^{4} d^{4}}{a b^{5} + b^{6} x} + \frac {e^{4} x^{3}}{3 b^{2}} - \frac {4 e \left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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