Optimal. Leaf size=97 \[ \frac {(b d-a e)^4 \log (a+b x)}{b^5}+\frac {e x (b d-a e)^3}{b^4}+\frac {(d+e x)^2 (b d-a e)^2}{2 b^3}+\frac {(d+e x)^3 (b d-a e)}{3 b^2}+\frac {(d+e x)^4}{4 b} \]
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Rubi [A] time = 0.04, antiderivative size = 97, 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 x (b d-a e)^3}{b^4}+\frac {(d+e x)^2 (b d-a e)^2}{2 b^3}+\frac {(d+e x)^3 (b d-a e)}{3 b^2}+\frac {(b d-a e)^4 \log (a+b x)}{b^5}+\frac {(d+e x)^4}{4 b} \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}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^4}{a+b x} \, dx\\ &=\int \left (\frac {e (b d-a e)^3}{b^4}+\frac {(b d-a e)^4}{b^4 (a+b x)}+\frac {e (b d-a e)^2 (d+e x)}{b^3}+\frac {e (b d-a e) (d+e x)^2}{b^2}+\frac {e (d+e x)^3}{b}\right ) \, dx\\ &=\frac {e (b d-a e)^3 x}{b^4}+\frac {(b d-a e)^2 (d+e x)^2}{2 b^3}+\frac {(b d-a e) (d+e x)^3}{3 b^2}+\frac {(d+e x)^4}{4 b}+\frac {(b d-a e)^4 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 114, normalized size = 1.18 \begin {gather*} \frac {b e x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)}{12 b^5} \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}{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 [A] time = 0.39, size = 181, normalized size = 1.87 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} + 4 \, {\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (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}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \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 = 1.79 \begin {gather*} \frac {3 \, b^{3} x^{4} e^{4} + 16 \, b^{3} d x^{3} e^{3} + 36 \, b^{3} d^{2} x^{2} e^{2} + 48 \, b^{3} d^{3} x e - 4 \, a b^{2} x^{3} e^{4} - 24 \, a b^{2} d x^{2} e^{3} - 72 \, a b^{2} d^{2} x e^{2} + 6 \, a^{2} b x^{2} e^{4} + 48 \, a^{2} b d x e^{3} - 12 \, a^{3} x e^{4}}{12 \, b^{4}} + \frac {{\left (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}\right )} \log \left ({\left | b x + a \right |}\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 = 209, normalized size = 2.15 \begin {gather*} \frac {e^{4} x^{4}}{4 b}-\frac {a \,e^{4} x^{3}}{3 b^{2}}+\frac {4 d \,e^{3} x^{3}}{3 b}+\frac {a^{2} e^{4} x^{2}}{2 b^{3}}-\frac {2 a d \,e^{3} x^{2}}{b^{2}}+\frac {3 d^{2} e^{2} x^{2}}{b}+\frac {a^{4} e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} d \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {a^{3} e^{4} x}{b^{4}}+\frac {6 a^{2} d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {4 a^{2} d \,e^{3} x}{b^{3}}-\frac {4 a \,d^{3} e \ln \left (b x +a \right )}{b^{2}}-\frac {6 a \,d^{2} e^{2} x}{b^{2}}+\frac {d^{4} \ln \left (b x +a \right )}{b}+\frac {4 d^{3} e x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 179, normalized size = 1.85 \begin {gather*} \frac {3 \, b^{3} e^{4} x^{4} + 4 \, {\left (4 \, b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{3} d^{3} e - 6 \, a b^{2} d^{2} e^{2} + 4 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} x}{12 \, b^{4}} + \frac {{\left (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}\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 = 0.05, size = 188, normalized size = 1.94 \begin {gather*} x\,\left (\frac {4\,d^3\,e}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,e^4}{b^2}-\frac {4\,d\,e^3}{b}\right )}{b}+\frac {6\,d^2\,e^2}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,e^4}{3\,b^2}-\frac {4\,d\,e^3}{3\,b}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,e^4}{b^2}-\frac {4\,d\,e^3}{b}\right )}{2\,b}+\frac {3\,d^2\,e^2}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (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\right )}{b^5}+\frac {e^4\,x^4}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 136, normalized size = 1.40 \begin {gather*} x^{3} \left (- \frac {a e^{4}}{3 b^{2}} + \frac {4 d e^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{2} e^{4}}{2 b^{3}} - \frac {2 a d e^{3}}{b^{2}} + \frac {3 d^{2} e^{2}}{b}\right ) + x \left (- \frac {a^{3} e^{4}}{b^{4}} + \frac {4 a^{2} d e^{3}}{b^{3}} - \frac {6 a d^{2} e^{2}}{b^{2}} + \frac {4 d^{3} e}{b}\right ) + \frac {e^{4} x^{4}}{4 b} + \frac {\left (a e - b d\right )^{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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