Optimal. Leaf size=143 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4} \]
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Rubi [A] time = 0.116061, antiderivative size = 143, 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 = {446, 96, 94, 93, 208} \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 96
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2}}{x^5 \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3 \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4}-\frac{\left (\frac{b c}{2}+\frac{3 a d}{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2 \sqrt{c+d x}} \, dx,x,x^2\right )}{4 a c}\\ &=\frac{(b c+3 a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4}-\frac{((b c-a d) (b c+3 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{16 a c^2}\\ &=\frac{(b c+3 a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4}-\frac{((b c-a d) (b c+3 a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}}\right )}{8 a c^2}\\ &=\frac{(b c+3 a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4}+\frac{(b c-a d) (b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0740333, size = 125, normalized size = 0.87 \[ \frac{\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (-2 a c+3 a d x^2-b c x^2\right )}{8 a c^2 x^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 355, normalized size = 2.5 \begin{align*} -{\frac{1}{16\,a{c}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 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 ){x}^{4}{a}^{2}{d}^{2}-2\,\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 ){x}^{4}abcd-\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac \right ) } \right ){x}^{4}{b}^{2}{c}^{2}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}da{x}^{2}\sqrt{ac}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}bc{x}^{2}\sqrt{ac}+4\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}ac\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+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 = 3.4225, size = 795, normalized size = 5.56 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{a c} x^{4} \log \left (\frac{{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{x^{4}}\right ) + 4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{32 \, a^{2} c^{3} x^{4}}, -\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{-a c} x^{4} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{2 \,{\left (a b c d x^{4} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{16 \, a^{2} c^{3} x^{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{\sqrt{a + b x^{2}}}{x^{5} \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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