Optimal. Leaf size=177 \[ \frac {16 d^4 (d g+e f)^2}{e^3 (d-e x)}+\frac {32 d^3 (d g+e f) (2 d g+e f) \log (d-e x)}{e^3}+\frac {1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac {d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac {d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac {1}{2} e g x^4 (3 d g+e f)+\frac {1}{5} e^2 g^2 x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {848, 88} \[ \frac {1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac {d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac {d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac {16 d^4 (d g+e f)^2}{e^3 (d-e x)}+\frac {32 d^3 (d g+e f) (2 d g+e f) \log (d-e x)}{e^3}+\frac {1}{2} e g x^4 (3 d g+e f)+\frac {1}{5} e^2 g^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^6 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^4 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {d^2 \left (17 e^2 f^2+64 d e f g+48 d^2 g^2\right )}{e^2}+\frac {2 d \left (3 e^2 f^2+17 d e f g+16 d^2 g^2\right ) x}{e}+\left (e^2 f^2+12 d e f g+17 d^2 g^2\right ) x^2+2 e g (e f+3 d g) x^3+e^2 g^2 x^4+\frac {32 d^3 (-e f-2 d g) (e f+d g)}{e^2 (d-e x)}+\frac {16 d^4 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {d^2 \left (17 e^2 f^2+64 d e f g+48 d^2 g^2\right ) x}{e^2}+\frac {d \left (3 e^2 f^2+17 d e f g+16 d^2 g^2\right ) x^2}{e}+\frac {1}{3} \left (e^2 f^2+12 d e f g+17 d^2 g^2\right ) x^3+\frac {1}{2} e g (e f+3 d g) x^4+\frac {1}{5} e^2 g^2 x^5+\frac {16 d^4 (e f+d g)^2}{e^3 (d-e x)}+\frac {32 d^3 (e f+d g) (e f+2 d g) \log (d-e x)}{e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 185, normalized size = 1.05 \[ -\frac {16 d^4 (d g+e f)^2}{e^3 (e x-d)}+\frac {1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac {d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac {d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac {32 d^3 \left (2 d^2 g^2+3 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {1}{2} e g x^4 (3 d g+e f)+\frac {1}{5} e^2 g^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 288, normalized size = 1.63 \[ \frac {6 \, e^{6} g^{2} x^{6} - 480 \, d^{4} e^{2} f^{2} - 960 \, d^{5} e f g - 480 \, d^{6} g^{2} + 3 \, {\left (5 \, e^{6} f g + 13 \, d e^{5} g^{2}\right )} x^{5} + 5 \, {\left (2 \, e^{6} f^{2} + 21 \, d e^{5} f g + 25 \, d^{2} e^{4} g^{2}\right )} x^{4} + 10 \, {\left (8 \, d e^{5} f^{2} + 39 \, d^{2} e^{4} f g + 31 \, d^{3} e^{3} g^{2}\right )} x^{3} + 30 \, {\left (14 \, d^{2} e^{4} f^{2} + 47 \, d^{3} e^{3} f g + 32 \, d^{4} e^{2} g^{2}\right )} x^{2} - 30 \, {\left (17 \, d^{3} e^{3} f^{2} + 64 \, d^{4} e^{2} f g + 48 \, d^{5} e g^{2}\right )} x - 960 \, {\left (d^{4} e^{2} f^{2} + 3 \, d^{5} e f g + 2 \, d^{6} g^{2} - {\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{30 \, {\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 327, normalized size = 1.85 \[ 16 \, {\left (2 \, d^{5} g^{2} e^{5} + 3 \, d^{4} f g e^{6} + d^{3} f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{30} \, {\left (6 \, g^{2} x^{5} e^{22} + 45 \, d g^{2} x^{4} e^{21} + 170 \, d^{2} g^{2} x^{3} e^{20} + 480 \, d^{3} g^{2} x^{2} e^{19} + 1440 \, d^{4} g^{2} x e^{18} + 15 \, f g x^{4} e^{22} + 120 \, d f g x^{3} e^{21} + 510 \, d^{2} f g x^{2} e^{20} + 1920 \, d^{3} f g x e^{19} + 10 \, f^{2} x^{3} e^{22} + 90 \, d f^{2} x^{2} e^{21} + 510 \, d^{2} f^{2} x e^{20}\right )} e^{\left (-20\right )} + \frac {16 \, {\left (2 \, d^{6} g^{2} e^{6} + 3 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {16 \, {\left (d^{7} g^{2} e^{5} + 2 \, d^{6} f g e^{6} + d^{5} f^{2} e^{7} + {\left (d^{6} g^{2} e^{6} + 2 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 245, normalized size = 1.38 \[ \frac {e^{2} g^{2} x^{5}}{5}+\frac {3 d e \,g^{2} x^{4}}{2}+\frac {e^{2} f g \,x^{4}}{2}+\frac {17 d^{2} g^{2} x^{3}}{3}+4 d e f g \,x^{3}+\frac {e^{2} f^{2} x^{3}}{3}+\frac {16 d^{3} g^{2} x^{2}}{e}+17 d^{2} f g \,x^{2}+3 d e \,f^{2} x^{2}-\frac {16 d^{6} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {32 d^{5} f g}{\left (e x -d \right ) e^{2}}+\frac {64 d^{5} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {16 d^{4} f^{2}}{\left (e x -d \right ) e}+\frac {96 d^{4} f g \ln \left (e x -d \right )}{e^{2}}+\frac {48 d^{4} g^{2} x}{e^{2}}+\frac {32 d^{3} f^{2} \ln \left (e x -d \right )}{e}+\frac {64 d^{3} f g x}{e}+17 d^{2} f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 218, normalized size = 1.23 \[ -\frac {16 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {6 \, e^{4} g^{2} x^{5} + 15 \, {\left (e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{4} + 10 \, {\left (e^{4} f^{2} + 12 \, d e^{3} f g + 17 \, d^{2} e^{2} g^{2}\right )} x^{3} + 30 \, {\left (3 \, d e^{3} f^{2} + 17 \, d^{2} e^{2} f g + 16 \, d^{3} e g^{2}\right )} x^{2} + 30 \, {\left (17 \, d^{2} e^{2} f^{2} + 64 \, d^{3} e f g + 48 \, d^{4} g^{2}\right )} x}{30 \, e^{2}} + \frac {32 \, {\left (d^{3} e^{2} f^{2} + 3 \, d^{4} e f g + 2 \, d^{5} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.61, size = 565, normalized size = 3.19 \[ x^2\,\left (\frac {2\,d\,\left (d^2\,g^2+3\,d\,e\,f\,g+e^2\,f^2\right )}{e}-\frac {d^2\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{2\,e^2}+\frac {d\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{e^2}-d^2\,g^2+\frac {2\,d\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{e}\right )}{e}\right )+x^4\,\left (\frac {e\,g\,\left (2\,d\,g+e\,f\right )}{2}+\frac {d\,e\,g^2}{2}\right )+x\,\left (\frac {d^4\,g^2+8\,d^3\,e\,f\,g+6\,d^2\,e^2\,f^2}{e^2}-\frac {d^2\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{e^2}-d^2\,g^2+\frac {2\,d\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{e}\right )}{e^2}+\frac {2\,d\,\left (\frac {4\,d\,\left (d^2\,g^2+3\,d\,e\,f\,g+e^2\,f^2\right )}{e}-\frac {d^2\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{e^2}+\frac {2\,d\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{e^2}-d^2\,g^2+\frac {2\,d\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{e}\right )}{e}\right )}{e}\right )+x^3\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{3\,e^2}-\frac {d^2\,g^2}{3}+\frac {2\,d\,\left (2\,e\,g\,\left (2\,d\,g+e\,f\right )+2\,d\,e\,g^2\right )}{3\,e}\right )+\frac {\ln \left (e\,x-d\right )\,\left (64\,d^5\,g^2+96\,d^4\,e\,f\,g+32\,d^3\,e^2\,f^2\right )}{e^3}+\frac {16\,\left (d^6\,g^2+2\,d^5\,e\,f\,g+d^4\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {e^2\,g^2\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.01, size = 199, normalized size = 1.12 \[ \frac {32 d^{3} \left (d g + e f\right ) \left (2 d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} + \frac {e^{2} g^{2} x^{5}}{5} + x^{4} \left (\frac {3 d e g^{2}}{2} + \frac {e^{2} f g}{2}\right ) + x^{3} \left (\frac {17 d^{2} g^{2}}{3} + 4 d e f g + \frac {e^{2} f^{2}}{3}\right ) + x^{2} \left (\frac {16 d^{3} g^{2}}{e} + 17 d^{2} f g + 3 d e f^{2}\right ) + x \left (\frac {48 d^{4} g^{2}}{e^{2}} + \frac {64 d^{3} f g}{e} + 17 d^{2} f^{2}\right ) + \frac {- 16 d^{6} g^{2} - 32 d^{5} e f g - 16 d^{4} e^{2} f^{2}}{- d e^{3} + e^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________