Optimal. Leaf size=366 \[ -\frac{e^2 \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{e^{3/2} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{16 a^{3/2} c^{9/2}}+\frac{d^2 e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac{e^2 (b c-a d)^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{e^3 (11 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
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Rubi [A] time = 0.368982, antiderivative size = 366, 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} \[ -\frac{e^2 \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{e^{3/2} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{16 a^{3/2} c^{9/2}}+\frac{d^2 e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac{e^2 (b c-a d)^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{e^3 (11 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
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 \frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{x^4 \left (b e-d x^2\right )^2}{\left (-a e+c x^2\right )^4} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{x^4 \left (-6 b^2 c^2 e^2+5 (b c e-a d e)^2+6 a c d^2 e 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 )}{6 a c^2}\\ &=\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(b c-a d)^2 (b c+11 a d) e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{a c (b c-a d) (b c+11 a d) e^3+4 c^2 (b c-a d) (b c+11 a d) e^2 x^2-24 a c^3 d^2 e 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 )}{24 a c^5}\\ &=\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(b c-a d)^2 (b c+11 a d) e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \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{3 a c \left (b^2 c^2+10 a b c d-19 a^2 d^2\right ) e^3-48 a^2 c^2 d^2 e^2 x^2}{-a e+c x^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )}{48 a^2 c^5 e}\\ &=\frac{d^2 (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(b c-a d)^2 (b c+11 a d) e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) 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 )}{16 a c^4}\\ &=\frac{d^2 (b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(b c-a d)^2 (b c+11 a d) e^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) 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 )}{16 a^{3/2} c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.19232, size = 245, normalized size = 0.67 \[ \frac{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (3 x^6 \sqrt{c+d x^2} \left (-45 a^2 b c d^2+35 a^3 d^3+9 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )-\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \left (a^2 \left (-14 c^2 d x^2+8 c^3+35 c d^2 x^4+105 d^3 x^6\right )+2 a b c x^2 \left (7 c^2-19 c d x^2-50 d^2 x^4\right )+3 b^2 c^2 x^4 \left (c+d x^2\right )\right )\right )}{48 a^{3/2} c^{9/2} x^6 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 1498, 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 = 93.0066, size = 1211, normalized size = 3.31 \begin{align*} \left [\frac{3 \,{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \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 (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e +{\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{192 \, a c^{4} x^{6}}, -\frac{3 \,{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \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 (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e +{\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{96 \, a c^{4} x^{6}}\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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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