Optimal. Leaf size=183 \[ \frac{(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt{d^2-e^2 x^2}}+\frac{(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{g^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]
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Rubi [A] time = 0.40349, antiderivative size = 183, normalized size of antiderivative = 1., 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 1635
Rule 778
Rule 217
Rule 203
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*}
Mathematica [A] time = 0.809311, size = 182, normalized size = 0.99 \[ \frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} (d g+e f) \left (d^2 e^2 \left (7 f^2+33 f g x+32 g^2 x^2\right )-d^3 e g (16 f+51 g x)+22 d^4 g^2-d e^3 f x (6 f+11 g x)+2 e^4 f^2 x^2\right )-15 d^2 g^3 (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{15 d^3 e^4 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.093, size = 713, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.57726, size = 1220, normalized size = 6.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21653, size = 917, normalized size = 5.01 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20298, size = 417, normalized size = 2.28 \begin{align*} -g^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )} \mathrm{sgn}\left (d\right ) - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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