Optimal. Leaf size=50 \[ -\frac {b (b d-a e) x}{e^2}+\frac {(a+b x)^2}{2 e}+\frac {(b d-a e)^2 \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45}
\begin {gather*} \frac {(b d-a e)^2 \log (d+e x)}{e^3}-\frac {b x (b d-a e)}{e^2}+\frac {(a+b x)^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx &=\int \frac {(a+b x)^2}{d+e x} \, dx\\ &=\int \left (-\frac {b (b d-a e)}{e^2}+\frac {b (a+b x)}{e}+\frac {(-b d+a e)^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {b (b d-a e) x}{e^2}+\frac {(a+b x)^2}{2 e}+\frac {(b d-a e)^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.86 \begin {gather*} \frac {b e x (-2 b d+4 a e+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 56, normalized size = 1.12
method | result | size |
default | \(\frac {b \left (\frac {1}{2} b e \,x^{2}+2 a e x -x b d \right )}{e^{2}}+\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(56\) |
norman | \(\frac {b \left (2 a e -b d \right ) x}{e^{2}}+\frac {b^{2} x^{2}}{2 e}+\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(59\) |
risch | \(\frac {b^{2} x^{2}}{2 e}+\frac {2 b a x}{e}-\frac {b^{2} x d}{e^{2}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}-\frac {2 \ln \left (e x +d \right ) a b d}{e^{2}}+\frac {\ln \left (e x +d \right ) b^{2} d^{2}}{e^{3}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 61, normalized size = 1.22 \begin {gather*} {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (b^{2} x^{2} e - 2 \, {\left (b^{2} d - 2 \, a b e\right )} x\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.07, size = 61, normalized size = 1.22 \begin {gather*} -\frac {1}{2} \, {\left (2 \, b^{2} d x e - {\left (b^{2} x^{2} + 4 \, a b x\right )} e^{2} - 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (x e + d\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 44, normalized size = 0.88 \begin {gather*} \frac {b^{2} x^{2}}{2 e} + x \left (\frac {2 a b}{e} - \frac {b^{2} d}{e^{2}}\right ) + \frac {\left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.03, size = 61, normalized size = 1.22 \begin {gather*} {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{2} x^{2} e - 2 \, b^{2} d x + 4 \, a b x e\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 62, normalized size = 1.24 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{e^3}-x\,\left (\frac {b^2\,d}{e^2}-\frac {2\,a\,b}{e}\right )+\frac {b^2\,x^2}{2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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