Optimal. Leaf size=183 \[ \frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {g^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}+\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.40, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1635, 778, 217, 203} \[ \frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {g^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 778
Rule 1635
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^3 f^3-9 d e^2 f^2 g-9 d^2 e f g^2-3 d^3 g^3}{e^3}+\frac {5 d g^2 (3 e f+d g) x}{e^2}+\frac {5 d g^3 x^2}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^3 f^3-9 d e^2 f^2 g+21 d^2 e f g^2+17 d^3 g^3}{e^3}+\frac {15 d^2 g^3 x}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^3}\\ &=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ &=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 182, normalized size = 0.99 \[ \frac {(d+e x) \left (\sqrt {1-\frac {e^2 x^2}{d^2}} (d g+e f) \left (22 d^4 g^2-d^3 e g (16 f+51 g x)+d^2 e^2 \left (7 f^2+33 f g x+32 g^2 x^2\right )-d e^3 f x (6 f+11 g x)+2 e^4 f^2 x^2\right )-15 d^2 g^3 (d-e x)^3 \sin ^{-1}\left (\frac {e x}{d}\right )\right )}{15 d^3 e^4 (d-e x)^2 \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 454, normalized size = 2.48 \[ -\frac {7 \, d^{3} e^{3} f^{3} - 9 \, d^{4} e^{2} f^{2} g + 6 \, d^{5} e f g^{2} + 22 \, d^{6} g^{3} - {\left (7 \, e^{6} f^{3} - 9 \, d e^{5} f^{2} g + 6 \, d^{2} e^{4} f g^{2} + 22 \, d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (7 \, d e^{5} f^{3} - 9 \, d^{2} e^{4} f^{2} g + 6 \, d^{3} e^{3} f g^{2} + 22 \, d^{4} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{4} f^{3} - 9 \, d^{3} e^{3} f^{2} g + 6 \, d^{4} e^{2} f g^{2} + 22 \, d^{5} e g^{3}\right )} x - 30 \, {\left (d^{3} e^{3} g^{3} x^{3} - 3 \, d^{4} e^{2} g^{3} x^{2} + 3 \, d^{5} e g^{3} x - d^{6} g^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (7 \, d^{2} e^{3} f^{3} - 9 \, d^{3} e^{2} f^{2} g + 6 \, d^{4} e f g^{2} + 22 \, d^{5} g^{3} + {\left (2 \, e^{5} f^{3} - 9 \, d e^{4} f^{2} g + 21 \, d^{2} e^{3} f g^{2} + 32 \, d^{3} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (2 \, d e^{4} f^{3} - 9 \, d^{2} e^{3} f^{2} g + 6 \, d^{3} e^{2} f g^{2} + 17 \, d^{4} e g^{3}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{7} x^{3} - 3 \, d^{4} e^{6} x^{2} + 3 \, d^{5} e^{5} x - d^{6} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 309, normalized size = 1.69 \[ -g^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\relax (d) - \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (x {\left (\frac {{\left (32 \, d^{4} g^{3} e^{8} + 21 \, d^{3} f g^{2} e^{9} - 9 \, d^{2} f^{2} g e^{10} + 2 \, d f^{3} e^{11}\right )} x e^{\left (-7\right )}}{d^{4}} + \frac {45 \, {\left (d^{5} g^{3} e^{7} + d^{4} f g^{2} e^{8}\right )} e^{\left (-7\right )}}{d^{4}}\right )} - \frac {5 \, {\left (7 \, d^{6} g^{3} e^{6} - 3 \, d^{5} f g^{2} e^{7} - 9 \, d^{4} f^{2} g e^{8} + d^{3} f^{3} e^{9}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x - \frac {5 \, {\left (11 \, d^{7} g^{3} e^{5} + 3 \, d^{6} f g^{2} e^{6} - 9 \, d^{5} f^{2} g e^{7} - d^{4} f^{3} e^{8}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x + \frac {15 \, {\left (d^{8} g^{3} e^{4} + d^{5} f^{3} e^{7}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x + \frac {{\left (22 \, d^{9} g^{3} e^{3} + 6 \, d^{8} f g^{2} e^{4} - 9 \, d^{7} f^{2} g e^{5} + 7 \, d^{6} f^{3} e^{6}\right )} e^{\left (-7\right )}}{d^{4}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 713, normalized size = 3.90 \[ \frac {e \,g^{3} x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 d \,g^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 e f \,g^{2} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 d^{2} g^{3} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {9 d f \,g^{2} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 e \,f^{2} g \,x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {11 d^{3} g^{3} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}-\frac {d^{2} f \,g^{2} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {3 d \,f^{2} g \,x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e \,f^{3} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {9 d^{4} g^{3} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {21 d^{3} f \,g^{2} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {9 d^{2} f^{2} g x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {4 d \,f^{3} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {g^{3} x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}+\frac {22 d^{5} g^{3}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}+\frac {2 d^{4} f \,g^{2}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {3 d^{3} f^{2} g}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}+\frac {7 d^{2} f^{3}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {3 d^{2} g^{3} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}+\frac {7 d f \,g^{2} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}+\frac {f^{3} x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d}-\frac {3 f^{2} g x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}+\frac {7 f \,g^{2} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{2}}-\frac {3 f^{2} g x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e}+\frac {2 f^{3} x}{15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}}+\frac {8 g^{3} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}-\frac {g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 891, normalized size = 4.87 \[ \frac {1}{15} \, e^{3} g^{3} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {1}{3} \, e g^{3} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {d f^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {3 \, d^{3} f^{2} g}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {4 \, f^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {4 \, d^{2} g^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} + \frac {8 \, f^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} - \frac {7 \, g^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}} + \frac {3 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {g^{3} \arcsin \left (\frac {e x}{d}\right )}{e^{4}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (e^{3} f^{3} + 9 \, d e^{2} f^{2} g + 9 \, d^{2} e f g^{2} + d^{3} g^{3}\right )} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {9 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {3 \, {\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {8 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} d^{4}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} - \frac {2 \, {\left (e^{3} f^{3} + 9 \, d e^{2} f^{2} g + 9 \, d^{2} e f g^{2} + d^{3} g^{3}\right )} d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {{\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{4}} - \frac {2 \, {\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, \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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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