Optimal. Leaf size=78 \[ -\frac {3 b^2 (b d-a e) \log (d+e x)}{e^4}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}+\frac {b^3 x}{e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 78, 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 {3 b^2 (b d-a e) \log (d+e x)}{e^4}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}+\frac {b^3 x}{e^3} \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)^3} \, dx &=\int \frac {(a+b x)^3}{(d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{e^3}+\frac {(-b d+a e)^3}{e^3 (d+e x)^3}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)^2}-\frac {3 b^2 (b d-a e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {b^3 x}{e^3}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}-\frac {3 b^2 (b d-a e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 114, normalized size = 1.46 \begin {gather*} \frac {-a^3 e^3-3 a^2 b e^2 (d+2 e x)+3 a b^2 d e (3 d+4 e x)-6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{2 e^4 (d+e x)^2} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 188, normalized size = 2.41 \begin {gather*} \frac {2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \, {\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} d^{3} - a b^{2} d^{2} e + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \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 = 1.41 \begin {gather*} b^{3} x e^{\left (-3\right )} - 3 \, {\left (b^{3} d - a b^{2} e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 160, normalized size = 2.05 \begin {gather*} -\frac {a^{3}}{2 \left (e x +d \right )^{2} e}+\frac {3 a^{2} b d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {3 a \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {b^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 a^{2} b}{\left (e x +d \right ) e^{2}}+\frac {6 a \,b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {3 a \,b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 125, normalized size = 1.60 \begin {gather*} \frac {b^{3} x}{e^{3}} - \frac {5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac {3 \, {\left (b^{3} d - a b^{2} e\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 = 2.04, size = 130, normalized size = 1.67 \begin {gather*} \frac {b^3\,x}{e^3}-\frac {\ln \left (d+e\,x\right )\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{e^4}-\frac {\frac {a^3\,e^3+3\,a^2\,b\,d\,e^2-9\,a\,b^2\,d^2\,e+5\,b^3\,d^3}{2\,e}+x\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 128, normalized size = 1.64 \begin {gather*} \frac {b^{3} x}{e^{3}} + \frac {3 b^{2} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a^{3} e^{3} - 3 a^{2} b d e^{2} + 9 a b^{2} d^{2} e - 5 b^{3} d^{3} + x \left (- 6 a^{2} b e^{3} + 12 a b^{2} d e^{2} - 6 b^{3} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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