Optimal. Leaf size=59 \[ -\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {2 b (b d-a e)}{e^3 (d+e x)}+\frac {b^2 \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, 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 {2 b (b d-a e)}{e^3 (d+e x)}-\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {b^2 \log (d+e x)}{e^3} \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)^3} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^3} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^3}-\frac {2 b (b d-a e)}{e^2 (d+e x)^2}+\frac {b^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac {2 b (b d-a e)}{e^3 (d+e x)}+\frac {b^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.81 \begin {gather*} \frac {\frac {(b d-a e) (3 b d+a e+4 b e x)}{(d+e x)^2}+2 b^2 \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 69, normalized size = 1.17
method | result | size |
risch | \(\frac {-\frac {2 b \left (a e -b d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}}{2 e^{3}}}{\left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(66\) |
norman | \(\frac {-\frac {a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}}{2 e^{3}}-\frac {2 \left (a b e -b^{2} d \right ) x}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(68\) |
default | \(-\frac {2 b \left (a e -b d \right )}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 77, normalized size = 1.31 \begin {gather*} b^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.68, size = 97, normalized size = 1.64 \begin {gather*} \frac {3 \, b^{2} d^{2} - {\left (4 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (2 \, b^{2} d x - a b d\right )} e + 2 \, {\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 80, normalized size = 1.36 \begin {gather*} \frac {b^{2} \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} - 2 a b d e + 3 b^{2} d^{2} + x \left (- 4 a b e^{2} + 4 b^{2} d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 69, normalized size = 1.17 \begin {gather*} b^{2} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (4 \, {\left (b^{2} d - a b e\right )} x + {\left (3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 77, normalized size = 1.31 \begin {gather*} \frac {b^2\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a^2\,e^2+2\,a\,b\,d\,e-3\,b^2\,d^2}{2\,e^3}+\frac {2\,b\,x\,\left (a\,e-b\,d\right )}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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