Optimal. Leaf size=118 \[ \frac {2 d^2 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {\left (13 d^2 g^2+10 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}-\frac {4 d (3 d g+e f) (d g+e f)}{e^3 (d-e x)}-\frac {g x (5 d g+2 e f)}{e^2}-\frac {g^2 x^2}{2 e} \]
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Rubi [A] time = 0.14, antiderivative size = 118, 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} \[ -\frac {\left (13 d^2 g^2+10 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {2 d^2 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {4 d (3 d g+e f) (d g+e f)}{e^3 (d-e x)}-\frac {g x (5 d g+2 e f)}{e^2}-\frac {g^2 x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^5 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^2 (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (-\frac {g (2 e f+5 d g)}{e^2}-\frac {g^2 x}{e}+\frac {4 d (-e f-3 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac {4 d^2 (e f+d g)^2}{e^2 (-d+e x)^3}+\frac {-e^2 f^2-10 d e f g-13 d^2 g^2}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {g (2 e f+5 d g) x}{e^2}-\frac {g^2 x^2}{2 e}+\frac {2 d^2 (e f+d g)^2}{e^3 (d-e x)^2}-\frac {4 d (e f+d g) (e f+3 d g)}{e^3 (d-e x)}-\frac {\left (e^2 f^2+10 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 118, normalized size = 1.00 \[ -\frac {\frac {8 d \left (3 d^2 g^2+4 d e f g+e^2 f^2\right )}{d-e x}+2 \left (13 d^2 g^2+10 d e f g+e^2 f^2\right ) \log (d-e x)-\frac {4 d^2 (d g+e f)^2}{(d-e x)^2}+2 e g x (5 d g+2 e f)+e^2 g^2 x^2}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 241, normalized size = 2.04 \[ -\frac {e^{4} g^{2} x^{4} + 4 \, d^{2} e^{2} f^{2} + 24 \, d^{3} e f g + 20 \, d^{4} g^{2} + 4 \, {\left (e^{4} f g + 2 \, d e^{3} g^{2}\right )} x^{3} - {\left (8 \, d e^{3} f g + 19 \, d^{2} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (4 \, d e^{3} f^{2} + 14 \, d^{2} e^{2} f g + 7 \, d^{3} e g^{2}\right )} x + 2 \, {\left (d^{2} e^{2} f^{2} + 10 \, d^{3} e f g + 13 \, d^{4} g^{2} + {\left (e^{4} f^{2} + 10 \, d e^{3} f g + 13 \, d^{2} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (d e^{3} f^{2} + 10 \, d^{2} e^{2} f g + 13 \, d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{2 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 273, normalized size = 2.31 \[ -\frac {1}{2} \, {\left (13 \, d^{2} g^{2} e^{5} + 10 \, d f g e^{6} + f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {1}{2} \, {\left (g^{2} x^{2} e^{11} + 10 \, d g^{2} x e^{10} + 4 \, f g x e^{11}\right )} e^{\left (-12\right )} - \frac {{\left (13 \, d^{3} g^{2} e^{4} + 10 \, d^{2} f g e^{5} + d f^{2} e^{6}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} - \frac {2 \, {\left (5 \, d^{6} g^{2} e^{5} + 6 \, d^{5} f g e^{6} + d^{4} f^{2} e^{7} - 2 \, {\left (3 \, d^{3} g^{2} e^{8} + 4 \, d^{2} f g e^{9} + d f^{2} e^{10}\right )} x^{3} - {\left (7 \, d^{4} g^{2} e^{7} + 10 \, d^{3} f g e^{8} + 3 \, d^{2} f^{2} e^{9}\right )} x^{2} + 4 \, {\left (d^{5} g^{2} e^{6} + d^{4} f g e^{7}\right )} x\right )} e^{\left (-8\right )}}{{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 198, normalized size = 1.68 \[ \frac {2 d^{4} g^{2}}{\left (e x -d \right )^{2} e^{3}}+\frac {4 d^{3} f g}{\left (e x -d \right )^{2} e^{2}}+\frac {2 d^{2} f^{2}}{\left (e x -d \right )^{2} e}-\frac {g^{2} x^{2}}{2 e}+\frac {12 d^{3} g^{2}}{\left (e x -d \right ) e^{3}}+\frac {16 d^{2} f g}{\left (e x -d \right ) e^{2}}-\frac {13 d^{2} g^{2} \ln \left (e x -d \right )}{e^{3}}+\frac {4 d \,f^{2}}{\left (e x -d \right ) e}-\frac {10 d f g \ln \left (e x -d \right )}{e^{2}}-\frac {5 d \,g^{2} x}{e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{e}-\frac {2 f g x}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 149, normalized size = 1.26 \[ -\frac {2 \, {\left (d^{2} e^{2} f^{2} + 6 \, d^{3} e f g + 5 \, d^{4} g^{2} - 2 \, {\left (d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac {e g^{2} x^{2} + 2 \, {\left (2 \, e f g + 5 \, d g^{2}\right )} x}{2 \, e^{2}} - \frac {{\left (e^{2} f^{2} + 10 \, d e f g + 13 \, d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 161, normalized size = 1.36 \[ -\frac {\frac {2\,\left (5\,d^4\,g^2+6\,d^3\,e\,f\,g+d^2\,e^2\,f^2\right )}{e}-x\,\left (12\,d^3\,g^2+16\,d^2\,e\,f\,g+4\,d\,e^2\,f^2\right )}{d^2\,e^2-2\,d\,e^3\,x+e^4\,x^2}-x\,\left (\frac {2\,g\,\left (d\,g+e\,f\right )}{e^2}+\frac {3\,d\,g^2}{e^2}\right )-\frac {\ln \left (e\,x-d\right )\,\left (13\,d^2\,g^2+10\,d\,e\,f\,g+e^2\,f^2\right )}{e^3}-\frac {g^2\,x^2}{2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 151, normalized size = 1.28 \[ - x \left (\frac {5 d g^{2}}{e^{2}} + \frac {2 f g}{e}\right ) - \frac {10 d^{4} g^{2} + 12 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x \left (- 12 d^{3} e g^{2} - 16 d^{2} e^{2} f g - 4 d e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} - \frac {g^{2} x^{2}}{2 e} - \frac {\left (13 d^{2} g^{2} + 10 d e f g + e^{2} f^{2}\right ) \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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