Optimal. Leaf size=50 \[ -\frac {g (e f+d g) x}{e^2}-\frac {(f+g x)^2}{2 e}-\frac {(e f+d g)^2 \log (d-e x)}{e^3} \]
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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 = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {813, 45}
\begin {gather*} -\frac {(d g+e f)^2 \log (d-e x)}{e^3}-\frac {g x (d g+e f)}{e^2}-\frac {(f+g x)^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 813
Rubi steps
\begin {align*} \int \frac {(d+e x) (f+g x)^2}{d^2-e^2 x^2} \, dx &=\int \frac {(f+g x)^2}{d-e x} \, dx\\ &=\int \left (-\frac {g (e f+d g)}{e^2}+\frac {(e f+d g)^2}{e^2 (d-e x)}-\frac {g (f+g x)}{e}\right ) \, dx\\ &=-\frac {g (e f+d g) x}{e^2}-\frac {(f+g x)^2}{2 e}-\frac {(e f+d g)^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 {e g x (4 e f+2 d g+e g x)+2 (e f+d g)^2 \log (d-e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 59, normalized size = 1.18
method | result | size |
default | \(-\frac {g \left (\frac {1}{2} e g \,x^{2}+d g x +2 e f x \right )}{e^{2}}+\frac {\left (-d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(59\) |
norman | \(-\frac {g^{2} x^{2}}{2 e}-\frac {g \left (d g +2 e f \right ) x}{e^{2}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(61\) |
risch | \(-\frac {g^{2} x^{2}}{2 e}-\frac {g^{2} d x}{e^{2}}-\frac {2 g f x}{e}-\frac {\ln \left (-e x +d \right ) d^{2} g^{2}}{e^{3}}-\frac {2 \ln \left (-e x +d \right ) d f g}{e^{2}}-\frac {\ln \left (-e x +d \right ) f^{2}}{e}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 1.28 \begin {gather*} -{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right ) - \frac {1}{2} \, {\left (g^{2} x^{2} e + 2 \, {\left (d g^{2} + 2 \, f g 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 = 2.06, size = 62, normalized size = 1.24 \begin {gather*} -\frac {1}{2} \, {\left (2 \, d g^{2} x e + {\left (g^{2} x^{2} + 4 \, f g x\right )} e^{2} + 2 \, {\left (d^{2} g^{2} + 2 \, d f g e + f^{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 = 0.11, size = 46, normalized size = 0.92 \begin {gather*} - x \left (\frac {d g^{2}}{e^{2}} + \frac {2 f g}{e}\right ) - \frac {g^{2} x^{2}}{2 e} - \frac {\left (d g + e f\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 = 1.35, size = 64, normalized size = 1.28 \begin {gather*} -{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right ) - \frac {1}{2} \, {\left (g^{2} x^{2} e + 2 \, d g^{2} x + 4 \, f g 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 = 2.61, size = 65, normalized size = 1.30 \begin {gather*} -x\,\left (\frac {d\,g^2}{e^2}+\frac {2\,f\,g}{e}\right )-\frac {\ln \left (e\,x-d\right )\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{e^3}-\frac {g^2\,x^2}{2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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