Optimal. Leaf size=94 \[ \frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 d} \]
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Rubi [A] time = 0.0659592, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {1591, 242, 47, 50, 63, 208} \[ \frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 242
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x \left (a+\frac{b}{c+d x^2}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+\frac{b}{x}\right )^{3/2} \, dx,x,c+d x^2\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,\frac{1}{c+d x^2}\right )}{2 d}\\ &=\frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{c+d x^2}\right )}{4 d}\\ &=-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}-\frac{(3 a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{c+d x^2}\right )}{4 d}\\ &=-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{c+d x^2}}\right )}{2 d}\\ &=-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{\left (c+d x^2\right ) \left (a+\frac{b}{c+d x^2}\right )^{3/2}}{2 d}+\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.102966, size = 79, normalized size = 0.84 \[ \frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (a \left (c+d x^2\right )-2 b\right )+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 336, normalized size = 3.6 \begin{align*}{\frac{1}{4\,d} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd}{\sqrt{a{d}^{2}}}} \right ){x}^{2}ab{d}^{2}+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}{x}^{2}ad+3\,\ln \left ( 1/2\,{\frac{2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd}{\sqrt{a{d}^{2}}}} \right ) abcd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}ac-4\,\sqrt{a{d}^{2}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }b \right ) \sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}{\frac{1}{\sqrt{a{d}^{2}}}}} \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 = 2.20824, size = 618, normalized size = 6.57 \begin{align*} \left [\frac{3 \, \sqrt{a} b \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \,{\left (2 \, a d^{2} x^{4} +{\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt{a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \,{\left (a d x^{2} + a c - 2 \, b\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, d}, -\frac{3 \, \sqrt{-a} b \arctan \left (\frac{{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt{-a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \,{\left (a d x^{2} + a c - 2 \, b\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, d}\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 x \left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.37699, size = 510, normalized size = 5.43 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 \, \sqrt{a} b{\left | d \right |} \log \left (12 \,{\left | a \right |}^{\frac{3}{2}}{\left | d \right |}\right )}{d^{2}} + \frac{\sqrt{a} b \log \left ({\left | -2 \, a^{\frac{5}{2}} c^{3} d - 6 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} c^{2}{\left | d \right |} - 6 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac{3}{2}} c d - a^{\frac{3}{2}} b c^{2} d - 2 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a{\left | d \right |} - 2 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b c{\left | d \right |} -{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} \sqrt{a} b d \right |}\right )}{{\left | d \right |}} - \frac{2 \, \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} a}{d}\right )} \mathrm{sgn}\left (d x^{2} + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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