Optimal. Leaf size=50 \[ -\frac {(d g+e f)^2 \log (d-e x)}{e^3}-\frac {g x (d g+e f)}{e^2}-\frac {(f+g x)^2}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, 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 = {799, 43} \[ -\frac {g x (d g+e f)}{e^2}-\frac {(d g+e f)^2 \log (d-e x)}{e^3}-\frac {(f+g x)^2}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 799
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 43, normalized size = 0.86 \[ -\frac {e g x (2 d g+4 e f+e g x)+2 (d g+e f)^2 \log (d-e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 64, normalized size = 1.28 \[ -\frac {e^{2} g^{2} x^{2} + 2 \, {\left (2 \, e^{2} f g + d e g^{2}\right )} x + 2 \, {\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 134, normalized size = 2.68 \[ -\frac {1}{2} \, {\left (d^{2} g^{2} e + 2 \, d f g e^{2} + f^{2} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {1}{2} \, {\left (g^{2} x^{2} e^{3} + 2 \, d g^{2} x e^{2} + 4 \, f g x e^{3}\right )} e^{\left (-4\right )} - \frac {{\left (d^{3} g^{2} + 2 \, d^{2} f g e + d f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 82, normalized size = 1.64 \[ -\frac {g^{2} x^{2}}{2 e}-\frac {d^{2} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {2 d f g \ln \left (e x -d \right )}{e^{2}}-\frac {d \,g^{2} x}{e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{e}-\frac {2 f g x}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 63, normalized size = 1.26 \[ -\frac {e g^{2} x^{2} + 2 \, {\left (2 \, e f g + d g^{2}\right )} x}{2 \, e^{2}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.61, size = 65, normalized size = 1.30 \[ -x\,\left (\frac {d\,g^2}{e^2}+\frac {2\,f\,g}{e}\right )-\frac {\ln \left (e\,x-d\right )\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{e^3}-\frac {g^2\,x^2}{2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.29, size = 46, normalized size = 0.92 \[ - x \left (\frac {d g^{2}}{e^{2}} + \frac {2 f g}{e}\right ) - \frac {g^{2} x^{2}}{2 e} - \frac {\left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________