Optimal. Leaf size=68 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]
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Rubi [A] time = 0.0639165, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 80, 63, 208} \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{c+d x^2}}{b d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\sqrt{c+d x^2}}{b d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{b d}\\ &=\frac{\sqrt{c+d x^2}}{b d}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.0540831, size = 68, normalized size = 1. \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2}}{b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 318, normalized size = 4.7 \begin{align*}{\frac{1}{bd}\sqrt{d{x}^{2}+c}}+{\frac{a}{2\,{b}^{2}}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{a}{2\,{b}^{2}}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \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 = 1.5331, size = 648, normalized size = 9.53 \begin{align*} \left [\frac{\sqrt{b^{2} c - a b d} a d \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}{4 \,{\left (b^{3} c d - a b^{2} d^{2}\right )}}, \frac{\sqrt{-b^{2} c + a b d} a d \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}{2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}{2 \,{\left (b^{3} c d - a b^{2} d^{2}\right )}}\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{x^{3}}{\left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12499, size = 86, normalized size = 1.26 \begin{align*} -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x^{2} + c}}{b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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