Optimal. Leaf size=75 \[ -\frac {b^2 x (2 b d-3 a e)}{e^3}+\frac {(b d-a e)^3}{e^4 (d+e x)}+\frac {3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac {b^3 x^2}{2 e^2} \]
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Rubi [A] time = 0.06, antiderivative size = 75, 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} \[ -\frac {b^2 x (2 b d-3 a e)}{e^3}+\frac {(b d-a e)^3}{e^4 (d+e x)}+\frac {3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac {b^3 x^2}{2 e^2} \]
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)^2} \, dx &=\int \frac {(a+b x)^3}{(d+e x)^2} \, dx\\ &=\int \left (-\frac {b^2 (2 b d-3 a e)}{e^3}+\frac {b^3 x}{e^2}+\frac {(-b d+a e)^3}{e^3 (d+e x)^2}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {b^2 (2 b d-3 a e) x}{e^3}+\frac {b^3 x^2}{2 e^2}+\frac {(b d-a e)^3}{e^4 (d+e x)}+\frac {3 b (b d-a e)^2 \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 114, normalized size = 1.52 \[ \frac {3 \left (a^2 b e^2-2 a b^2 d e+b^3 d^2\right ) \log (d+e x)}{e^4}+\frac {-a^3 e^3+3 a^2 b d e^2-3 a b^2 d^2 e+b^3 d^3}{e^4 (d+e x)}-\frac {b^2 x (2 b d-3 a e)}{e^3}+\frac {b^3 x^2}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 172, normalized size = 2.29 \[ \frac {b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \, {\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \, {\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 164, normalized size = 2.19 \[ \frac {1}{2} \, {\left (b^{3} - \frac {6 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} e^{\left (-4\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {b^{3} d^{3} e^{2}}{x e + d} - \frac {3 \, a b^{2} d^{2} e^{3}}{x e + d} + \frac {3 \, a^{2} b d e^{4}}{x e + d} - \frac {a^{3} e^{5}}{x e + d}\right )} e^{\left (-6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 149, normalized size = 1.99 \[ \frac {b^{3} x^{2}}{2 e^{2}}-\frac {a^{3}}{\left (e x +d \right ) e}+\frac {3 a^{2} b d}{\left (e x +d \right ) e^{2}}+\frac {3 a^{2} b \ln \left (e x +d \right )}{e^{2}}-\frac {3 a \,b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 a \,b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {3 a \,b^{2} x}{e^{2}}+\frac {b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 b^{3} d x}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 117, normalized size = 1.56 \[ \frac {b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{e^{5} x + d e^{4}} + \frac {b^{3} e x^{2} - 2 \, {\left (2 \, b^{3} d - 3 \, a b^{2} e\right )} x}{2 \, e^{3}} + \frac {3 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 123, normalized size = 1.64 \[ x\,\left (\frac {3\,a\,b^2}{e^2}-\frac {2\,b^3\,d}{e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}{e^4}-\frac {a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}{e\,\left (x\,e^4+d\,e^3\right )}+\frac {b^3\,x^2}{2\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 102, normalized size = 1.36 \[ \frac {b^{3} x^{2}}{2 e^{2}} + \frac {3 b \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{4}} + x \left (\frac {3 a b^{2}}{e^{2}} - \frac {2 b^{3} d}{e^{3}}\right ) + \frac {- a^{3} e^{3} + 3 a^{2} b d e^{2} - 3 a b^{2} d^{2} e + b^{3} d^{3}}{d e^{4} + e^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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