Optimal. Leaf size=260 \[ -\frac{x (b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}+\frac{b x \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}+\frac{a \sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{d} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{(b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.291417, antiderivative size = 348, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6722, 1974, 413, 531, 418, 492, 411} \[ -\frac{x (b-a c) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{a \left (c+d x^2\right )+b}}+\frac{b x \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{a \left (c+d x^2\right )+b}}+\frac{a \sqrt{c} \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{d} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{(b-a c) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1974
Rule 413
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \left (a+\frac{b}{c+d x^2}\right )^{3/2} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\left (b+a \left (c+d x^2\right )\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, 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{\left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \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{a c (b+a c) d-a (b-a c) d^2 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c d \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (a (b+a c) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (a (b-a c) d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{b+a \left (c+d x^2\right )}}-\frac{(b-a c) x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{b+a \left (c+d x^2\right )}}+\frac{a \sqrt{c} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{d} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left ((b-a c) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{b+a \left (c+d x^2\right )}}-\frac{(b-a c) x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c \sqrt{b+a \left (c+d x^2\right )}}+\frac{(b-a c) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}+\frac{a \sqrt{c} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{d} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}
Mathematica [C] time = 0.542941, size = 243, normalized size = 0.93 \[ \frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (b x \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )-2 i a b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-i a c (a c-b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{c \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 515, normalized size = 2. \begin{align*}{\frac{1}{c \left ( ad{x}^{2}+ac+b \right ) } \left ( \sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{-{\frac{ad}{ac+b}}}{x}^{3}abd+\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }{a}^{2}{c}^{2}+2\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }abc-\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }abc+\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{-{\frac{ad}{ac+b}}}xabc+\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{-{\frac{ad}{ac+b}}}x{b}^{2} \right ) \sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}}{\frac{1}{\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}}}{\frac{1}{\sqrt{-{\frac{ad}{ac+b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a d x^{2} + a c + b\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{d x^{2} + c}, x\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 \left (a + \frac{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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