Optimal. Leaf size=73 \[ \frac {(b d-a e)^3 \log (a+b x)}{b^4}+\frac {e x (b d-a e)^2}{b^3}+\frac {(d+e x)^2 (b d-a e)}{2 b^2}+\frac {(d+e x)^3}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 73, 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)^2}{b^3}+\frac {(d+e x)^2 (b d-a e)}{2 b^2}+\frac {(b d-a e)^3 \log (a+b x)}{b^4}+\frac {(d+e x)^3}{3 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)^3}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^3}{a+b x} \, dx\\ &=\int \left (\frac {e (b d-a e)^2}{b^3}+\frac {(b d-a e)^3}{b^3 (a+b x)}+\frac {e (b d-a e) (d+e x)}{b^2}+\frac {e (d+e x)^2}{b}\right ) \, dx\\ &=\frac {e (b d-a e)^2 x}{b^3}+\frac {(b d-a e) (d+e x)^2}{2 b^2}+\frac {(d+e x)^3}{3 b}+\frac {(b d-a e)^3 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 1.01 \begin {gather*} \frac {b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)}{6 b^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)^3}{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 = 0.38, size = 116, normalized size = 1.59 \begin {gather*} \frac {2 \, b^{3} e^{3} x^{3} + 3 \, {\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 110, normalized size = 1.51 \begin {gather*} \frac {2 \, b^{2} x^{3} e^{3} + 9 \, b^{2} d x^{2} e^{2} + 18 \, b^{2} d^{2} x e - 3 \, a b x^{2} e^{3} - 18 \, a b d x e^{2} + 6 \, a^{2} x e^{3}}{6 \, b^{3}} + \frac {{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 133, normalized size = 1.82 \begin {gather*} \frac {e^{3} x^{3}}{3 b}-\frac {a \,e^{3} x^{2}}{2 b^{2}}+\frac {3 d \,e^{2} x^{2}}{2 b}-\frac {a^{3} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 a^{2} d \,e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {a^{2} e^{3} x}{b^{3}}-\frac {3 a \,d^{2} e \ln \left (b x +a \right )}{b^{2}}-\frac {3 a d \,e^{2} x}{b^{2}}+\frac {d^{3} \ln \left (b x +a \right )}{b}+\frac {3 d^{2} e x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 114, normalized size = 1.56 \begin {gather*} \frac {2 \, b^{2} e^{3} x^{3} + 3 \, {\left (3 \, b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x}{6 \, b^{3}} + \frac {{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 118, normalized size = 1.62 \begin {gather*} x\,\left (\frac {3\,d^2\,e}{b}+\frac {a\,\left (\frac {a\,e^3}{b^2}-\frac {3\,d\,e^2}{b}\right )}{b}\right )-x^2\,\left (\frac {a\,e^3}{2\,b^2}-\frac {3\,d\,e^2}{2\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{b^4}+\frac {e^3\,x^3}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 83, normalized size = 1.14 \begin {gather*} x^{2} \left (- \frac {a e^{3}}{2 b^{2}} + \frac {3 d e^{2}}{2 b}\right ) + x \left (\frac {a^{2} e^{3}}{b^{3}} - \frac {3 a d e^{2}}{b^{2}} + \frac {3 d^{2} e}{b}\right ) + \frac {e^{3} x^{3}}{3 b} - \frac {\left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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