Optimal. Leaf size=86 \[ \frac {(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}-\frac {(e f-3 d g) (d g+e f) \log (d-e x)}{4 d^2 e^3}+\frac {(d g+e f)^2}{2 d e^3 (d-e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 86, 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, 88} \[ \frac {(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}-\frac {(e f-3 d g) (d g+e f) \log (d-e x)}{4 d^2 e^3}+\frac {(d g+e f)^2}{2 d e^3 (d-e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 799
Rubi steps
\begin {align*} \int \frac {(d+e x) (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{2 d e^2 (d-e x)^2}+\frac {(e f-3 d g) (e f+d g)}{4 d^2 e^2 (d-e x)}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)}\right ) \, dx\\ &=\frac {(e f+d g)^2}{2 d e^3 (d-e x)}-\frac {(e f-3 d g) (e f+d g) \log (d-e x)}{4 d^2 e^3}+\frac {(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 91, normalized size = 1.06 \[ \frac {(d-e x) \left (3 d^2 g^2+2 d e f g-e^2 f^2\right ) \log (d-e x)+(d-e x) (e f-d g)^2 \log (d+e x)+2 d (d g+e f)^2}{4 d^2 e^3 (d-e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.55, size = 168, normalized size = 1.95 \[ -\frac {2 \, d e^{2} f^{2} + 4 \, d^{2} e f g + 2 \, d^{3} g^{2} + {\left (d e^{2} f^{2} - 2 \, d^{2} e f g + d^{3} g^{2} - {\left (e^{3} f^{2} - 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) - {\left (d e^{2} f^{2} - 2 \, d^{2} e f g - 3 \, d^{3} g^{2} - {\left (e^{3} f^{2} - 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \, {\left (d^{2} e^{4} x - d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 159, normalized size = 1.85 \[ \frac {1}{2} \, g^{2} e^{\left (-3\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {{\left (d^{2} g^{2} + 2 \, d f g e - f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{4 \, d {\left | d \right |}} - \frac {{\left ({\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} x + {\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3}\right )} e^{\left (-2\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x^{2} e^{2} - d^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 156, normalized size = 1.81 \[ -\frac {d \,g^{2}}{2 \left (e x -d \right ) e^{3}}-\frac {f^{2}}{2 \left (e x -d \right ) d e}+\frac {f g \ln \left (e x -d \right )}{2 d \,e^{2}}-\frac {f g \ln \left (e x +d \right )}{2 d \,e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{4 d^{2} e}+\frac {f^{2} \ln \left (e x +d \right )}{4 d^{2} e}-\frac {f g}{\left (e x -d \right ) e^{2}}+\frac {3 g^{2} \ln \left (e x -d \right )}{4 e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{4 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 114, normalized size = 1.33 \[ -\frac {e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}}{2 \, {\left (d e^{4} x - d^{2} e^{3}\right )}} + \frac {{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac {{\left (e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.64, size = 111, normalized size = 1.29 \[ \frac {d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2}{2\,d\,e^3\,\left (d-e\,x\right )}+\frac {\ln \left (d+e\,x\right )\,\left (d^2\,g^2-2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^2\,e^3}+\frac {\ln \left (d-e\,x\right )\,\left (3\,d^2\,g^2+2\,d\,e\,f\,g-e^2\,f^2\right )}{4\,d^2\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.04, size = 182, normalized size = 2.12 \[ \frac {- d^{2} g^{2} - 2 d e f g - e^{2} f^{2}}{- 2 d^{2} e^{3} + 2 d e^{4} x} + \frac {\left (d g - e f\right )^{2} \log {\left (x + \frac {2 d^{3} g^{2} - d \left (d g - e f\right )^{2}}{d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} + \frac {\left (d g + e f\right ) \left (3 d g - e f\right ) \log {\left (x + \frac {2 d^{3} g^{2} - d \left (d g + e f\right ) \left (3 d g - e f\right )}{d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________