Optimal. Leaf size=282 \[ -\frac{e^{3/2} (b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{16 b^{3/2} d^{9/2}}+\frac{e \left (c+d x^2\right ) \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 b d^4}+\frac{c^2 e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{d^4}+\frac{\left (c+d x^2\right )^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 b d^2 e}-\frac{e \left (c+d x^2\right )^2 (a d+11 b c) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 d^4} \]
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Rubi [A] time = 0.383243, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1960, 463, 455, 1157, 388, 208} \[ -\frac{e^{3/2} (b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{16 b^{3/2} d^{9/2}}+\frac{e \left (c+d x^2\right ) \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 b d^4}+\frac{c^2 e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{d^4}+\frac{\left (c+d x^2\right )^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 b d^2 e}-\frac{e \left (c+d x^2\right )^2 (a d+11 b c) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 d^4} \]
Antiderivative was successfully verified.
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Rule 1960
Rule 463
Rule 455
Rule 1157
Rule 388
Rule 208
Rubi steps
\begin{align*} \int x^5 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{x^4 \left (-a e+c x^2\right )^2}{\left (b e-d x^2\right )^4} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{x^4 \left (-6 a^2 d^2 e^2+5 (b c e-a d e)^2+6 b c^2 d e x^2\right )}{\left (b e-d x^2\right )^3} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 b d^2}\\ &=-\frac{(11 b c+a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 d^4}+\frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{-b d (b c-a d) (11 b c+a d) e^3-4 d^2 (b c-a d) (11 b c+a d) e^2 x^2-24 b c^2 d^3 e x^4}{\left (b e-d x^2\right )^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{24 b d^5}\\ &=\frac{\left (79 b^2 c^2-50 a b c d-5 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{48 b d^4}-\frac{(11 b c+a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 d^4}+\frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2} \left (c+d x^2\right )^3}{6 b d^2 e}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{-3 b d \left (19 b^2 c^2-10 a b c d-a^2 d^2\right ) e^3-48 b^2 c^2 d^2 e^2 x^2}{b e-d x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{48 b^2 d^5 e}\\ &=\frac{c^2 (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{d^4}+\frac{\left (79 b^2 c^2-50 a b c d-5 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{48 b d^4}-\frac{(11 b c+a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 d^4}+\frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac{\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{b e-d x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{16 b d^4}\\ &=\frac{c^2 (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{d^4}+\frac{\left (79 b^2 c^2-50 a b c d-5 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{48 b d^4}-\frac{(11 b c+a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 d^4}+\frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac{(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right )}{16 b^{3/2} d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.569633, size = 294, normalized size = 1.04 \[ \frac{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (b \sqrt{d} \sqrt{b c-a d} \left (a^2 b d \left (-100 c^2-35 c d x^2+17 d^2 x^4\right )+3 a^3 d^2 \left (c+d x^2\right )+a b^2 \left (-65 c^2 d x^2+105 c^3-52 c d^2 x^4+22 d^3 x^6\right )+b^3 x^2 \left (35 c^2 d x^2+105 c^3-14 c d^2 x^4+8 d^3 x^6\right )\right )-3 \sqrt{a+b x^2} (b c-a d)^2 \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )\right )}{48 b^2 d^{9/2} \left (a+b x^2\right ) \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 1027, normalized size = 3.6 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 17.1062, size = 1175, normalized size = 4.17 \begin{align*} \left [\frac{3 \,{\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e \sqrt{\frac{e}{b d}} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} e x^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e - 4 \,{\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d + a b c d^{2} +{\left (3 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{\frac{e}{b d}}\right ) + 4 \,{\left (8 \, b^{2} d^{3} e x^{6} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e x^{4} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} e x^{2} +{\left (105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{192 \, b d^{4}}, \frac{3 \,{\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e \sqrt{-\frac{e}{b d}} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{-\frac{e}{b d}}}{2 \,{\left (b e x^{2} + a e\right )}}\right ) + 2 \,{\left (8 \, b^{2} d^{3} e x^{6} - 14 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e x^{4} +{\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} e x^{2} +{\left (105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{96 \, b d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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