Optimal. Leaf size=122 \[ \frac {(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {799, 88, 208} \[ \frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac {(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 208
Rule 799
Rubi steps
\begin {align*} \int \frac {(d+e x) (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^3 (d+e x)^2} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{4 d^2 e^2 (d-e x)^3}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{8 d^3 e^2 (d+e x)^2}+\frac {(e f-d g) (3 e f+d g)}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{8 d^2 e^3 (d-e x)^2}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {((e f-d g) (3 e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=\frac {(e f+d g)^2}{8 d^2 e^3 (d-e x)^2}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac {(e f-d g) (3 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 140, normalized size = 1.15 \[ \frac {\frac {4 d e^2 f^2-4 d^3 g^2}{d-e x}+\left (d^2 g^2+2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2-2 d e f g+3 e^2 f^2\right ) \log (d+e x)+\frac {2 d^2 (d g+e f)^2}{(d-e x)^2}-\frac {2 d (e f-d g)^2}{d+e x}}{16 d^4 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.07, size = 417, normalized size = 3.42 \[ \frac {4 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \, {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x + {\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} + {\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - {\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) - {\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} + {\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - {\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{16 \, {\left (d^{4} e^{6} x^{3} - d^{5} e^{5} x^{2} - d^{6} e^{4} x + d^{7} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 191, normalized size = 1.57 \[ \frac {{\left (d^{2} g^{2} + 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{16 \, d^{3} {\left | d \right |}} + \frac {{\left (d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} + d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 2 \, d^{3} f g x e^{3} + 4 \, d^{4} f g e^{2} - 3 \, f^{2} x^{3} e^{6} + 5 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{8 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 257, normalized size = 2.11 \[ \frac {f g}{4 \left (e x -d \right )^{2} d \,e^{2}}+\frac {f^{2}}{8 \left (e x -d \right )^{2} d^{2} e}+\frac {g^{2}}{8 \left (e x -d \right )^{2} e^{3}}+\frac {g^{2}}{4 \left (e x -d \right ) d \,e^{3}}-\frac {g^{2}}{8 \left (e x +d \right ) d \,e^{3}}+\frac {f g}{4 \left (e x +d \right ) d^{2} e^{2}}+\frac {g^{2} \ln \left (e x -d \right )}{16 d^{2} e^{3}}-\frac {g^{2} \ln \left (e x +d \right )}{16 d^{2} e^{3}}-\frac {f^{2}}{4 \left (e x -d \right ) d^{3} e}-\frac {f^{2}}{8 \left (e x +d \right ) d^{3} e}+\frac {f g \ln \left (e x -d \right )}{8 d^{3} e^{2}}-\frac {f g \ln \left (e x +d \right )}{8 d^{3} e^{2}}-\frac {3 f^{2} \ln \left (e x -d \right )}{16 d^{4} e}+\frac {3 f^{2} \ln \left (e x +d \right )}{16 d^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 211, normalized size = 1.73 \[ \frac {2 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} - {\left (3 \, e^{4} f^{2} - 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} + {\left (3 \, d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \, {\left (d^{3} e^{6} x^{3} - d^{4} e^{5} x^{2} - d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac {{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac {{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.64, size = 198, normalized size = 1.62 \[ \frac {\frac {-d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2}{4\,d\,e^3}+\frac {x\,\left (3\,d^2\,g^2-2\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+2\,d\,e\,f\,g-3\,e^2\,f^2\right )}{8\,d^3\,e}}{d^3-d^2\,e\,x-d\,e^2\,x^2+e^3\,x^3}-\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g-e\,f\right )\,\left (d\,g+3\,e\,f\right )}{d\,\left (d^2\,g^2+2\,d\,e\,f\,g-3\,e^2\,f^2\right )}\right )\,\left (d\,g-e\,f\right )\,\left (d\,g+3\,e\,f\right )}{8\,d^4\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.32, size = 277, normalized size = 2.27 \[ - \frac {2 d^{4} g^{2} - 4 d^{3} e f g - 2 d^{2} e^{2} f^{2} + x^{2} \left (- d^{2} e^{2} g^{2} - 2 d e^{3} f g + 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} + 2 d^{2} e^{2} f g - 3 d e^{3} f^{2}\right )}{8 d^{6} e^{3} - 8 d^{5} e^{4} x - 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac {\left (d g - e f\right ) \left (d g + 3 e f\right ) \log {\left (- \frac {d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac {\left (d g - e f\right ) \left (d g + 3 e f\right ) \log {\left (\frac {d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________