Optimal. Leaf size=137 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (a d+3 b c)}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d} \]
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Rubi [A] time = 0.1272, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {446, 80, 50, 63, 217, 206} \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (a d+3 b c)}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3 \sqrt{a+b x^2}}{\sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x \sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d}-\frac{(3 b c+a d) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{8 b d}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d}+\frac{((b c-a d) (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{16 b d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d}+\frac{((b c-a d) (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x^2}\right )}{8 b^2 d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d}+\frac{((b c-a d) (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}}\right )}{8 b^2 d^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{8 b d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 b d}+\frac{(b c-a d) (3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{3/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.299721, size = 138, normalized size = 1.01 \[ \frac{b \sqrt{d} \sqrt{a+b x^2} \left (c+d x^2\right ) \left (a d-3 b c+2 b d x^2\right )+(a d+3 b c) (b c-a d)^{3/2} \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )}{8 b^2 d^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 339, normalized size = 2.5 \begin{align*} -{\frac{1}{16\,{d}^{2}b}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( -4\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}bd+{d}^{2}\ln \left ({\frac{1}{2} \left ( 2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cabd-3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}-2\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}ad+6\,\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}bc \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}}}{\frac{1}{\sqrt{bd}}}} \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 = 1.93929, size = 752, normalized size = 5.49 \begin{align*} \left [-\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d}\right ) - 4 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{32 \, b^{2} d^{3}}, -\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d}}{2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{16 \, b^{2} d^{3}}\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} \sqrt{a + b x^{2}}}{\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 = 1.21667, size = 207, normalized size = 1.51 \begin{align*} \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (\frac{2 \,{\left (b x^{2} + a\right )}}{b d} - \frac{3 \, b^{2} c d + a b d^{2}}{b^{2} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \log \left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}}{8 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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