Optimal. Leaf size=256 \[ -\frac{a e^3 (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{e^2 (5 b c-9 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 e^{3/2} (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{8 \sqrt{a} c^{7/2}}-\frac{d e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^3} \]
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Rubi [A] time = 0.217923, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1960, 455, 1157, 388, 208} \[ -\frac{a e^3 (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{e^2 (5 b c-9 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 e^{3/2} (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{8 \sqrt{a} c^{7/2}}-\frac{d e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^3} \]
Antiderivative was successfully verified.
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Rule 1960
Rule 455
Rule 1157
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{x^4 \left (b e-d x^2\right )}{\left (-a e+c x^2\right )^3} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac{a (b c-a d)^2 e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{((b c-a d) e) \operatorname{Subst}\left (\int \frac{-a (b c-a d) e^2-4 c (b c-a d) e x^2+4 c^2 d x^4}{\left (-a e+c x^2\right )^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 c^3}\\ &=-\frac{a (b c-a d)^2 e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{(5 b c-9 a d) (b c-a d) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{-a (3 b c-7 a d) e^2+8 a c d e x^2}{-a e+c x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 a c^3}\\ &=-\frac{d (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac{a (b c-a d)^2 e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{(5 b c-9 a d) (b c-a d) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{\left (3 (b c-5 a d) (b c-a d) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a e+c x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 c^3}\\ &=-\frac{d (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac{a (b c-a d)^2 e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{(5 b c-9 a d) (b c-a d) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 (b c-5 a d) (b c-a d) e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{8 \sqrt{a} c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.124734, size = 186, normalized size = 0.73 \[ -\frac{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (3 x^4 \sqrt{c+d x^2} \left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \left (a \left (2 c^2-5 c d x^2-15 d^2 x^4\right )+b c x^2 \left (5 c+13 d x^2\right )\right )\right )}{8 \sqrt{a} c^{7/2} x^4 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 1042, normalized size = 4.1 \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 = 19.6949, size = 925, normalized size = 3.61 \begin{align*} \left [\frac{3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt{\frac{e}{a c}} \log \left (\frac{{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \,{\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \,{\left (2 \, a^{2} c^{3} +{\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} +{\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{\frac{e}{a c}}}{x^{4}}\right ) - 4 \,{\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \,{\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{32 \, c^{3} x^{4}}, \frac{3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} e x^{4} \sqrt{-\frac{e}{a c}} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}} \sqrt{-\frac{e}{a c}}}{2 \,{\left (b e x^{2} + a e\right )}}\right ) - 2 \,{\left ({\left (13 \, b c d - 15 \, a d^{2}\right )} e x^{4} + 2 \, a c^{2} e + 5 \,{\left (b c^{2} - a c d\right )} e x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{16 \, c^{3} x^{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 \frac{\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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