Optimal. Leaf size=62 \[ -\frac {c d \left (c d^2-a e^2\right ) x}{e^2}+\frac {(a e+c d x)^2}{2 e}+\frac {\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} -\frac {c d x \left (c d^2-a e^2\right )}{e^2}+\frac {\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}+\frac {(a e+c d x)^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^3} \, dx &=\int \frac {(a e+c d x)^2}{d+e x} \, dx\\ &=\int \left (-\frac {c d \left (c d^2-a e^2\right )}{e^2}+\frac {c d (a e+c d x)}{e}+\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {c d \left (c d^2-a e^2\right ) x}{e^2}+\frac {(a e+c d x)^2}{2 e}+\frac {\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 52, normalized size = 0.84 \begin {gather*} \frac {c d e x \left (4 a e^2+c d (-2 d+e x)\right )+2 \left (c d^2-a e^2\right )^2 \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 66, normalized size = 1.06
method | result | size |
default | \(\frac {c d \left (\frac {1}{2} c d e \,x^{2}+2 a \,e^{2} x -c \,d^{2} x \right )}{e^{2}}+\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(66\) |
risch | \(\frac {c^{2} d^{2} x^{2}}{2 e}+2 c d a x -\frac {c^{2} d^{3} x}{e^{2}}+e \ln \left (e x +d \right ) a^{2}-\frac {2 \ln \left (e x +d \right ) a c \,d^{2}}{e}+\frac {\ln \left (e x +d \right ) c^{2} d^{4}}{e^{3}}\) | \(77\) |
norman | \(\frac {-\frac {d^{2} \left (8 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right )}{2 e^{3}}-\frac {2 d \left (3 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) x}{e^{2}}+\frac {e \,c^{2} d^{2} x^{4}}{2}+2 x^{3} a d \,e^{2} c}{\left (e x +d \right )^{2}}+\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 69, normalized size = 1.11 \begin {gather*} {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c^{2} d^{2} x^{2} e - 2 \, {\left (c^{2} d^{3} - 2 \, a c d e^{2}\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 = 2.79, size = 68, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, {\left (c^{2} d^{2} x^{2} e^{2} - 2 \, c^{2} d^{3} x e + 4 \, a c d x e^{3} + 2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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 = 0.13, size = 53, normalized size = 0.85 \begin {gather*} \frac {c^{2} d^{2} x^{2}}{2 e} + x \left (2 a c d - \frac {c^{2} d^{3}}{e^{2}}\right ) + \frac {\left (a e^{2} - c d^{2}\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 = 1.46, size = 69, normalized size = 1.11 \begin {gather*} {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c^{2} d^{2} x^{2} e - 2 \, c^{2} d^{3} x + 4 \, a c d x e^{2}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 69, normalized size = 1.11 \begin {gather*} x\,\left (2\,a\,c\,d-\frac {c^2\,d^3}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{e^3}+\frac {c^2\,d^2\,x^2}{2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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