Optimal. Leaf size=104 \[ \frac{b d \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right )}{2 c^{3/2} \sqrt{a c+b}}-\frac{\left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{2 c x^2} \]
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Rubi [A] time = 0.385875, antiderivative size = 140, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6722, 1975, 446, 94, 93, 208} \[ \frac{b d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a \left (c+d x^2\right )+b}}\right )}{2 c^{3/2} \sqrt{a c+b} \sqrt{a \left (c+d x^2\right )+b}}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 c x^2} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 446
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^3} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a \left (c+d x^2\right )}}{x^3 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{x^3 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x}}{x^2 \sqrt{c+d x}} \, dx,x,x^2\right )}{2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 c x^2}-\frac{\left (b d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{4 c \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 c x^2}-\frac{\left (b d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{-c-(-b-a c) x^2} \, dx,x,\frac{\sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{2 c \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 c x^2}+\frac{b d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{b+a c} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{b+a \left (c+d x^2\right )}}\right )}{2 c^{3/2} \sqrt{b+a c} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}
Mathematica [A] time = 0.356297, size = 132, normalized size = 1.27 \[ \frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\frac{b d \sqrt{c+d x^2} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a c+a d x^2+b}}\right )}{\sqrt{a c+b} \sqrt{a \left (c+d x^2\right )+b}}-\frac{\sqrt{c} \left (c+d x^2\right )}{x^2}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 454, normalized size = 4.4 \begin{align*} -{\frac{d{x}^{2}+c}{4\,{c}^{2} \left ( ac+b \right ){x}^{2}}\sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}} \left ( -2\,a{d}^{2}\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}{x}^{4}\sqrt{{c}^{2}a+bc}-\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,acd{x}^{2}+bd{x}^{2}+2\,{c}^{2}a+2\,\sqrt{{c}^{2}a+bc}\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}+2\,bc \right ) } \right ){x}^{2}ab{c}^{2}d-4\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}acd{x}^{2}\sqrt{{c}^{2}a+bc}-\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,acd{x}^{2}+bd{x}^{2}+2\,{c}^{2}a+2\,\sqrt{{c}^{2}a+bc}\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}+2\,bc \right ) } \right ){x}^{2}{b}^{2}cd-2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}bd{x}^{2}\sqrt{{c}^{2}a+bc}+2\, \left ( a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc \right ) ^{3/2}\sqrt{{c}^{2}a+bc} \right ){\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}{\frac{1}{\sqrt{{c}^{2}a+bc}}}} \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 [B] time = 2.1593, size = 936, normalized size = 9. \begin{align*} \left [\frac{\sqrt{a c^{2} + b c} b d x^{2} \log \left (\frac{{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \,{\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \,{\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} +{\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt{a c^{2} + b c} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \,{\left (a c^{3} +{\left (a c^{2} + b c\right )} d x^{2} + b c^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{8 \,{\left (a c^{3} + b c^{2}\right )} x^{2}}, -\frac{\sqrt{-a c^{2} - b c} b d x^{2} \arctan \left (\frac{{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt{-a c^{2} - b c} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} c^{3} + 2 \, a b c^{2} +{\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \,{\left (a c^{3} +{\left (a c^{2} + b c\right )} d x^{2} + b c^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{4 \,{\left (a c^{3} + b c^{2}\right )} x^{2}}\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{\frac{a c + a d x^{2} + b}{c + d x^{2}}}}{x^{3}}\, dx \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{a + \frac{b}{d x^{2} + c}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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