Optimal. Leaf size=204 \[ \frac{c^{3/2} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.0925536, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {422, 418, 492, 411} \[ \frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx &=c \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx+d \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\\ &=\frac{d x \sqrt{a+b x^2}}{b \sqrt{c+d x^2}}+\frac{c^{3/2} \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{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}{b}\\ &=\frac{d x \sqrt{a+b x^2}}{b \sqrt{c+d 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 )}{b \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{c^{3/2} \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{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 = 0.0496046, size = 86, normalized size = 0.42 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 101, normalized size = 0.5 \begin{align*}{\frac{c}{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}\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 ){\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{\sqrt{d x^{2} + c}}{\sqrt{b x^{2} + a}}\,{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{d x^{2} + c}}{\sqrt{b x^{2} + a}}, 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{\sqrt{c + d x^{2}}}{\sqrt{a + b 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{\sqrt{d x^{2} + c}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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