Optimal. Leaf size=179 \[ -\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac {(e f+2 d g) (e f+12 d g) x^2}{2 e}-\frac {1}{3} g (2 e f+7 d g) x^3-\frac {1}{4} e g^2 x^4+\frac {8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}-\frac {32 d^3 (e f+d g) (e f+2 d g)}{e^3 (d-e x)}-\frac {8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 179, 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 = {862, 90}
\begin {gather*} \frac {8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac {d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac {8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac {1}{3} g x^3 (7 d g+2 e f)-\frac {x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac {1}{4} e g^2 x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 862
Rubi steps
\begin {align*} \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^4 (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (-\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right )}{e^2}+\frac {(-e f-12 d g) (e f+2 d g) x}{e}-g (2 e f+7 d g) x^2-e g^2 x^3+\frac {32 d^3 (-e f-2 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac {16 d^4 (e f+d g)^2}{e^2 (-d+e x)^3}-\frac {8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right )}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac {(e f+2 d g) (e f+12 d g) x^2}{2 e}-\frac {1}{3} g (2 e f+7 d g) x^3-\frac {1}{4} e g^2 x^4+\frac {8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}-\frac {32 d^3 (e f+d g) (e f+2 d g)}{e^3 (d-e x)}-\frac {8 d^2 \left (3 e^2 f^2+14 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 = 0.06, size = 193, normalized size = 1.08 \begin {gather*} -\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac {\left (e^2 f^2+14 d e f g+24 d^2 g^2\right ) x^2}{2 e}-\frac {1}{3} g (2 e f+7 d g) x^3-\frac {1}{4} e g^2 x^4+\frac {8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}+\frac {32 d^3 \left (e^2 f^2+3 d e f g+2 d^2 g^2\right )}{e^3 (-d+e x)}-\frac {8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 216, normalized size = 1.21
method | result | size |
default | \(-\frac {\frac {1}{4} g^{2} e^{3} x^{4}+\frac {7}{3} d \,e^{2} g^{2} x^{3}+\frac {2}{3} e^{3} f g \,x^{3}+12 d^{2} e \,g^{2} x^{2}+7 d \,e^{2} f g \,x^{2}+\frac {1}{2} e^{3} f^{2} x^{2}+56 d^{3} g^{2} x +48 d^{2} e f g x +7 d \,e^{2} f^{2} x}{e^{2}}-\frac {32 d^{3} \left (2 d^{2} g^{2}+3 d e f g +e^{2} f^{2}\right )}{e^{3} \left (-e x +d \right )}+\frac {8 d^{4} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right )}{e^{3} \left (-e x +d \right )^{2}}-\frac {8 d^{2} \left (13 d^{2} g^{2}+14 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(216\) |
risch | \(-\frac {e \,g^{2} x^{4}}{4}-\frac {7 x^{3} d \,g^{2}}{3}-\frac {2 e \,x^{3} f g}{3}-\frac {12 x^{2} d^{2} g^{2}}{e}-7 x^{2} d f g -\frac {e \,x^{2} f^{2}}{2}-\frac {56 d^{3} g^{2} x}{e^{2}}-\frac {48 d^{2} f g x}{e}-7 d \,f^{2} x +\frac {\left (64 d^{5} g^{2}+96 d^{4} e f g +32 d^{3} e^{2} f^{2}\right ) x -\frac {8 d^{4} \left (7 d^{2} g^{2}+10 d e f g +3 e^{2} f^{2}\right )}{e}}{e^{2} \left (-e x +d \right )^{2}}-\frac {104 d^{4} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {112 d^{3} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {24 d^{2} \ln \left (-e x +d \right ) f^{2}}{e}\) | \(216\) |
norman | \(\frac {\left (\frac {521}{3} d^{5} g^{2}+\frac {574}{3} d^{4} e f g +46 d^{3} e^{2} f^{2}\right ) x^{3}+\left (-\frac {154}{3} g^{2} d^{3} e^{2}-\frac {140}{3} f g \,e^{3} d^{2}-7 f^{2} d \,e^{4}\right ) x^{5}+\left (-\frac {23}{2} g^{2} e^{3} d^{2}-7 f g d \,e^{4}-\frac {1}{2} f^{2} e^{5}\right ) x^{6}-\frac {d^{4} \left (319 g^{2} e \,d^{4}+376 f g \,d^{3} e^{2}+100 f^{2} e^{3} d^{2}\right )}{4 e^{4}}-\frac {g^{2} e^{5} x^{8}}{4}+\frac {d^{2} \left (215 g^{2} e \,d^{4}+266 f g \,d^{3} e^{2}+83 f^{2} e^{3} d^{2}\right ) x^{2}}{2 e^{2}}-\frac {d^{5} \left (104 d^{2} g^{2}+112 d e f g +23 e^{2} f^{2}\right ) x}{e^{2}}-\frac {e^{4} g \left (7 d g +2 e f \right ) x^{7}}{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{2}}-\frac {8 d^{2} \left (13 d^{2} g^{2}+14 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(297\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 218, normalized size = 1.22 \begin {gather*} -8 \, {\left (13 \, d^{4} g^{2} + 14 \, d^{3} f g e + 3 \, d^{2} f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x e - d\right ) - \frac {1}{12} \, {\left (3 \, g^{2} x^{4} e^{3} + 4 \, {\left (7 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x^{3} + 6 \, {\left (24 \, d^{2} g^{2} e + 14 \, d f g e^{2} + f^{2} e^{3}\right )} x^{2} + 12 \, {\left (56 \, d^{3} g^{2} + 48 \, d^{2} f g e + 7 \, d f^{2} e^{2}\right )} x\right )} e^{\left (-2\right )} - \frac {8 \, {\left (7 \, d^{6} g^{2} + 10 \, d^{5} f g e + 3 \, d^{4} f^{2} e^{2} - 4 \, {\left (2 \, d^{5} g^{2} e + 3 \, d^{4} f g e^{2} + d^{3} f^{2} e^{3}\right )} x\right )}}{x^{2} e^{5} - 2 \, d x e^{4} + d^{2} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.08, size = 322, normalized size = 1.80 \begin {gather*} -\frac {672 \, d^{6} g^{2} + {\left (3 \, g^{2} x^{6} + 8 \, f g x^{5} + 6 \, f^{2} x^{4}\right )} e^{6} + 2 \, {\left (11 \, d g^{2} x^{5} + 34 \, d f g x^{4} + 36 \, d f^{2} x^{3}\right )} e^{5} + {\left (91 \, d^{2} g^{2} x^{4} + 416 \, d^{2} f g x^{3} - 162 \, d^{2} f^{2} x^{2}\right )} e^{4} + 4 \, {\left (103 \, d^{3} g^{2} x^{3} - 267 \, d^{3} f g x^{2} - 75 \, d^{3} f^{2} x\right )} e^{3} - 48 \, {\left (25 \, d^{4} g^{2} x^{2} + 12 \, d^{4} f g x - 6 \, d^{4} f^{2}\right )} e^{2} - 96 \, {\left (d^{5} g^{2} x - 10 \, d^{5} f g\right )} e + 96 \, {\left (13 \, d^{6} g^{2} + 3 \, d^{2} f^{2} x^{2} e^{4} + 2 \, {\left (7 \, d^{3} f g x^{2} - 3 \, d^{3} f^{2} x\right )} e^{3} + {\left (13 \, d^{4} g^{2} x^{2} - 28 \, d^{4} f g x + 3 \, d^{4} f^{2}\right )} e^{2} - 2 \, {\left (13 \, d^{5} g^{2} x - 7 \, d^{5} f g\right )} e\right )} \log \left (x e - d\right )}{12 \, {\left (x^{2} e^{5} - 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 219, normalized size = 1.22 \begin {gather*} - \frac {8 d^{2} \cdot \left (13 d^{2} g^{2} + 14 d e f g + 3 e^{2} f^{2}\right ) \log {\left (- d + e x \right )}}{e^{3}} - \frac {e g^{2} x^{4}}{4} - x^{3} \cdot \left (\frac {7 d g^{2}}{3} + \frac {2 e f g}{3}\right ) - x^{2} \cdot \left (\frac {12 d^{2} g^{2}}{e} + 7 d f g + \frac {e f^{2}}{2}\right ) - x \left (\frac {56 d^{3} g^{2}}{e^{2}} + \frac {48 d^{2} f g}{e} + 7 d f^{2}\right ) - \frac {56 d^{6} g^{2} + 80 d^{5} e f g + 24 d^{4} e^{2} f^{2} + x \left (- 64 d^{5} e g^{2} - 96 d^{4} e^{2} f g - 32 d^{3} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.80, size = 215, normalized size = 1.20 \begin {gather*} -8 \, {\left (13 \, d^{4} g^{2} + 14 \, d^{3} f g e + 3 \, d^{2} f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right ) - \frac {1}{12} \, {\left (3 \, g^{2} x^{4} e^{13} + 28 \, d g^{2} x^{3} e^{12} + 144 \, d^{2} g^{2} x^{2} e^{11} + 672 \, d^{3} g^{2} x e^{10} + 8 \, f g x^{3} e^{13} + 84 \, d f g x^{2} e^{12} + 576 \, d^{2} f g x e^{11} + 6 \, f^{2} x^{2} e^{13} + 84 \, d f^{2} x e^{12}\right )} e^{\left (-12\right )} - \frac {8 \, {\left (7 \, d^{6} g^{2} + 10 \, d^{5} f g e + 3 \, d^{4} f^{2} e^{2} - 4 \, {\left (2 \, d^{5} g^{2} e + 3 \, d^{4} f g e^{2} + d^{3} f^{2} e^{3}\right )} x\right )} e^{\left (-3\right )}}{{\left (x e - d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 375, normalized size = 2.09 \begin {gather*} \frac {x\,\left (64\,d^5\,g^2+96\,d^4\,e\,f\,g+32\,d^3\,e^2\,f^2\right )-\frac {8\,\left (7\,d^6\,g^2+10\,d^5\,e\,f\,g+3\,d^4\,e^2\,f^2\right )}{e}}{d^2\,e^2-2\,d\,e^3\,x+e^4\,x^2}-x^2\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{2\,e^3}-\frac {3\,d^2\,g^2}{2\,e}+\frac {3\,d\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{2\,e}\right )-x\,\left (\frac {d^3\,g^2}{e^2}-\frac {3\,d^2\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{e^2}+\frac {4\,d\,\left (d^2\,g^2+3\,d\,e\,f\,g+e^2\,f^2\right )}{e^2}+\frac {3\,d\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{e^3}-\frac {3\,d^2\,g^2}{e}+\frac {3\,d\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {2\,g\,\left (2\,d\,g+e\,f\right )}{3}+d\,g^2\right )-\frac {\ln \left (e\,x-d\right )\,\left (104\,d^4\,g^2+112\,d^3\,e\,f\,g+24\,d^2\,e^2\,f^2\right )}{e^3}-\frac {e\,g^2\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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