Optimal. Leaf size=74 \[ -\frac {(b d-a e)^3 \log (d+e x)}{e^4}+\frac {b x (b d-a e)^2}{e^3}-\frac {(a+b x)^2 (b d-a e)}{2 e^2}+\frac {(a+b x)^3}{3 e} \]
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} \frac {b x (b d-a e)^2}{e^3}-\frac {(a+b x)^2 (b d-a e)}{2 e^2}-\frac {(b d-a e)^3 \log (d+e x)}{e^4}+\frac {(a+b x)^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx &=\int \frac {(a+b x)^3}{d+e x} \, dx\\ &=\int \left (\frac {b (b d-a e)^2}{e^3}-\frac {b (b d-a e) (a+b x)}{e^2}+\frac {b (a+b x)^2}{e}+\frac {(-b d+a e)^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {b (b d-a e)^2 x}{e^3}-\frac {(b d-a e) (a+b x)^2}{2 e^2}+\frac {(a+b x)^3}{3 e}-\frac {(b d-a e)^3 \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 1.00 \begin {gather*} \frac {b e x \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)}{6 e^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) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 115, normalized size = 1.55 \begin {gather*} \frac {2 \, b^{3} e^{3} x^{3} - 3 \, {\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, 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 (e x + d\right )}{6 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 113, normalized size = 1.53 \begin {gather*} -{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, b^{3} x^{3} e^{2} - 3 \, b^{3} d x^{2} e + 6 \, b^{3} d^{2} x + 9 \, a b^{2} x^{2} e^{2} - 18 \, a b^{2} d x e + 18 \, a^{2} b x e^{2}\right )} e^{\left (-3\right )} \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 = 1.80 \begin {gather*} \frac {b^{3} x^{3}}{3 e}+\frac {3 a \,b^{2} x^{2}}{2 e}-\frac {b^{3} d \,x^{2}}{2 e^{2}}+\frac {a^{3} \ln \left (e x +d \right )}{e}-\frac {3 a^{2} b d \ln \left (e x +d \right )}{e^{2}}+\frac {3 a^{2} b x}{e}+\frac {3 a \,b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 a \,b^{2} d x}{e^{2}}-\frac {b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} d^{2} x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 114, normalized size = 1.54 \begin {gather*} \frac {2 \, b^{3} e^{2} x^{3} - 3 \, {\left (b^{3} d e - 3 \, a b^{2} e^{2}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} - 3 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x}{6 \, e^{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 (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.03, size = 118, normalized size = 1.59 \begin {gather*} x^2\,\left (\frac {3\,a\,b^2}{2\,e}-\frac {b^3\,d}{2\,e^2}\right )+x\,\left (\frac {3\,a^2\,b}{e}-\frac {d\,\left (\frac {3\,a\,b^2}{e}-\frac {b^3\,d}{e^2}\right )}{e}\right )+\frac {\ln \left (d+e\,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 )}{e^4}+\frac {b^3\,x^3}{3\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 83, normalized size = 1.12 \begin {gather*} \frac {b^{3} x^{3}}{3 e} + x^{2} \left (\frac {3 a b^{2}}{2 e} - \frac {b^{3} d}{2 e^{2}}\right ) + x \left (\frac {3 a^{2} b}{e} - \frac {3 a b^{2} d}{e^{2}} + \frac {b^{3} d^{2}}{e^{3}}\right ) + \frac {\left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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