Optimal. Leaf size=273 \[ -\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{a \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{c+d x^2}}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d x \sqrt{a+b x^2}}{a \sqrt{c+d x^2} (b c-a d)}+\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.143031, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {414, 21, 422, 418, 492, 411} \[ \frac{b x \sqrt{c+d x^2}}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d x \sqrt{a+b x^2}}{a \sqrt{c+d x^2} (b c-a d)}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c+d x^2} (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 414
Rule 21
Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx &=\frac{b x \sqrt{c+d x^2}}{a (b c-a d) \sqrt{a+b x^2}}-\frac{\int \frac{a d+b d x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{a (b c-a d)}\\ &=\frac{b x \sqrt{c+d x^2}}{a (b c-a d) \sqrt{a+b x^2}}-\frac{d \int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx}{a (b c-a d)}\\ &=\frac{b x \sqrt{c+d x^2}}{a (b c-a d) \sqrt{a+b x^2}}-\frac{d \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{b c-a d}-\frac{(b d) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{a (b c-a d)}\\ &=-\frac{d x \sqrt{a+b x^2}}{a (b c-a d) \sqrt{c+d x^2}}+\frac{b x \sqrt{c+d x^2}}{a (b c-a d) \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{(c d) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{a (b c-a d)}\\ &=-\frac{d x \sqrt{a+b x^2}}{a (b c-a d) \sqrt{c+d x^2}}+\frac{b x \sqrt{c+d x^2}}{a (b c-a d) \sqrt{a+b x^2}}+\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a (b c-a d) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.235924, size = 112, normalized size = 0.41 \[ \frac{\frac{a d \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}}}-b x \left (c+d x^2\right )}{a \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 248, normalized size = 0.9 \begin{align*}{\frac{1}{a \left ( ad-bc \right ) \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) } \left ( -{x}^{3}bd\sqrt{-{\frac{b}{a}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ad\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) bc\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) bc\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-xbc\sqrt{-{\frac{b}{a}}} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \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 \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c}}\,{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{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}}, 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 \frac{1}{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{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 \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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