Optimal. Leaf size=318 \[ -\frac{\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{\sqrt{e} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{16 a^{5/2} c^{7/2}}+\frac{e^2 (b c-a d)^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(3 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
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Rubi [A] time = 0.311522, antiderivative size = 318, 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, 463, 455, 385, 208} \[ -\frac{\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{\sqrt{e} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{16 a^{5/2} c^{7/2}}+\frac{e^2 (b c-a d)^3 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{(3 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
Antiderivative was successfully verified.
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Rule 1960
Rule 463
Rule 455
Rule 385
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{x^2 \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 )^{3/2}}{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^2 \left (-3 \left (2 b^2 c^2 e^2-(b c e-a d e)^2\right )+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)^2 (b c+3 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{(b c-a d)^3 e^2 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{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{3 c (b c-a d) (b c+3 a d) e^2-24 a c^2 d^2 e x^2}{\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^4}\\ &=\frac{(b c-a d)^2 (b c+3 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac{c \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 )^{3/2}}{6 a c^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac{\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) e\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^2 c^3}\\ &=\frac{(b c-a d)^2 (b c+3 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac{c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac{c \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 )^{3/2}}{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) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt{e} \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^{5/2} c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.17518, size = 222, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \sqrt{c+d x^2} \left (a^2 \left (-8 c^2+10 c d x^2-15 d^2 x^4\right )-2 a b c x^2 \left (c-2 d x^2\right )+3 b^2 c^2 x^4\right )-3 x^6 \left (3 a^2 b c d^2-5 a^3 d^3+a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )\right )}{48 a^{5/2} c^{7/2} x^6 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 849, normalized size = 2.7 \begin{align*} -{\frac{d{x}^{2}+c}{96\,{c}^{4}{a}^{3}{x}^{6}}\sqrt{{\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}}} \left ( -66\,b{d}^{3}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{8}{a}^{2}\sqrt{ac}-24\,{b}^{2}{d}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{8}ac\sqrt{ac}-6\,{b}^{3}d\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{8}{c}^{2}\sqrt{ac}-15\,{a}^{4}\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ){d}^{3}c{x}^{6}+9\,\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ){d}^{2}b{a}^{3}{c}^{2}{x}^{6}+3\,\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ) d{b}^{2}{a}^{2}{c}^{3}{x}^{6}+3\,{c}^{4}\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ){b}^{3}a{x}^{6}-66\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{d}^{3}{a}^{3}{x}^{6}\sqrt{ac}-54\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{d}^{2}b{a}^{2}c{x}^{6}\sqrt{ac}-18\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{b}^{2}da{c}^{2}{x}^{6}\sqrt{ac}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{b}^{3}{c}^{3}{x}^{6}\sqrt{ac}+66\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}{d}^{2}{a}^{2}{x}^{4}\sqrt{ac}+24\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}dbac{x}^{4}\sqrt{ac}+6\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}{b}^{2}{c}^{2}{x}^{4}\sqrt{ac}-36\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}d{a}^{2}c{x}^{2}\sqrt{ac}-12\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}ba{c}^{2}{x}^{2}\sqrt{ac}+16\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}{a}^{2}{c}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{ac}}}} \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 = 23.8737, size = 1162, normalized size = 3.65 \begin{align*} \left [-\frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} 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 + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} +{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{192 \, a^{2} c^{3} x^{6}}, \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} 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 + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} +{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \,{\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{96 \, a^{2} c^{3} 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{\sqrt{\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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