Optimal. Leaf size=145 \[ \frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1635, 789, 637} \[ \frac {(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}}+\frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 637
Rule 789
Rule 1635
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-2 f^2+\frac {6 d f g}{e}+\frac {3 d^2 g^2}{e^2}+\frac {5 d g^2 x}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2 e^2}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 110, normalized size = 0.76 \[ \frac {(d+e x) \left (2 d^4 g^2-6 d^3 e g (f+g x)+d^2 e^2 \left (7 f^2+18 f g x+7 g^2 x^2\right )-6 d e^3 f x (f+g x)+2 e^4 f^2 x^2\right )}{15 d^3 e^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 279, normalized size = 1.92 \[ -\frac {7 \, d^{3} e^{2} f^{2} - 6 \, d^{4} e f g + 2 \, d^{5} g^{2} - {\left (7 \, e^{5} f^{2} - 6 \, d e^{4} f g + 2 \, d^{2} e^{3} g^{2}\right )} x^{3} + 3 \, {\left (7 \, d e^{4} f^{2} - 6 \, d^{2} e^{3} f g + 2 \, d^{3} e^{2} g^{2}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{3} f^{2} - 6 \, d^{3} e^{2} f g + 2 \, d^{4} e g^{2}\right )} x + {\left (7 \, d^{2} e^{2} f^{2} - 6 \, d^{3} e f g + 2 \, d^{4} g^{2} + {\left (2 \, e^{4} f^{2} - 6 \, d e^{3} f g + 7 \, d^{2} e^{2} g^{2}\right )} x^{2} - 6 \, {\left (d e^{3} f^{2} - 3 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{6} x^{3} - 3 \, d^{4} e^{5} x^{2} + 3 \, d^{5} e^{4} x - d^{6} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 198, normalized size = 1.37 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left (15 \, d f^{2} + {\left ({\left ({\left (15 \, g^{2} e + \frac {{\left (7 \, d^{3} g^{2} e^{6} - 6 \, d^{2} f g e^{7} + 2 \, d f^{2} e^{8}\right )} x e^{\left (-4\right )}}{d^{4}}\right )} x + \frac {5 \, {\left (d^{5} g^{2} e^{4} + 6 \, d^{4} f g e^{5} - d^{3} f^{2} e^{6}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x - \frac {5 \, {\left (d^{6} g^{2} e^{3} - 6 \, d^{5} f g e^{4} - d^{4} f^{2} e^{5}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x\right )} x + \frac {{\left (2 \, d^{8} g^{2} e - 6 \, d^{7} f g e^{2} + 7 \, d^{6} f^{2} e^{3}\right )} e^{\left (-4\right )}}{d^{4}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 131, normalized size = 0.90 \[ \frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (7 d^{2} e^{2} g^{2} x^{2}-6 d \,e^{3} f g \,x^{2}+2 e^{4} f^{2} x^{2}-6 d^{3} e \,g^{2} x +18 d^{2} e^{2} f g x -6 d \,e^{3} f^{2} x +2 d^{4} g^{2}-6 d^{3} e f g +7 d^{2} e^{2} f^{2}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 583, normalized size = 4.02 \[ \frac {e g^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, d^{2} g^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d f^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{2}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {2 \, d^{3} f g}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {8 \, d^{4} g^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} + \frac {4 \, f^{2} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {8 \, f^{2} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {3 \, {\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{4}} - \frac {2 \, {\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.87, size = 125, normalized size = 0.86 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4\,g^2-6\,d^3\,e\,f\,g-6\,d^3\,e\,g^2\,x+7\,d^2\,e^2\,f^2+18\,d^2\,e^2\,f\,g\,x+7\,d^2\,e^2\,g^2\,x^2-6\,d\,e^3\,f^2\,x-6\,d\,e^3\,f\,g\,x^2+2\,e^4\,f^2\,x^2\right )}{15\,d^3\,e^3\,{\left (d-e\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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