Optimal. Leaf size=164 \[ \frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}{16 d^3}-\frac{5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)}{24 d^2}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{16 \sqrt{b} d^{7/2}}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d} \]
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Rubi [A] time = 0.140962, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {444, 50, 63, 217, 206} \[ \frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}{16 d^3}-\frac{5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)}{24 d^2}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{16 \sqrt{b} d^{7/2}}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d} \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}-\frac{(5 (b c-a d)) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{12 d}\\ &=-\frac{5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{24 d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}+\frac{\left (5 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{16 d^2}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{16 d^3}-\frac{5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{24 d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}-\frac{\left (5 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{32 d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{16 d^3}-\frac{5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{24 d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}-\frac{\left (5 (b c-a d)^3\right ) \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 )}{16 b d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{16 d^3}-\frac{5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{24 d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}-\frac{\left (5 (b c-a d)^3\right ) \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 )}{16 b d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{16 d^3}-\frac{5 (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{24 d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 d}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{16 \sqrt{b} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.514228, size = 164, normalized size = 1. \[ \frac{\sqrt{d} \sqrt{a+b x^2} \left (c+d x^2\right ) \left (33 a^2 d^2+2 a b d \left (13 d x^2-20 c\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )-\frac{15 (b c-a d)^{7/2} \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )}{b}}{48 d^{7/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 529, normalized size = 3.2 \begin{align*}{\frac{1}{96\,{d}^{3}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 16\,{x}^{4}{b}^{2}{d}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+52\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}ab{d}^{2}\sqrt{bd}-20\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}c{b}^{2}d\sqrt{bd}+15\,{d}^{3}\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 ){a}^{3}-45\,\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 ){a}^{2}cb{d}^{2}+45\,\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}a{b}^{2}d-15\,{b}^{3}\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}^{3}+66\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{a}^{2}{d}^{2}\sqrt{bd}-80\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}acbd\sqrt{bd}+30\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{c}^{2}{b}^{2}\sqrt{bd} \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 = 1.98681, size = 971, normalized size = 5.92 \begin{align*} \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{192 \, b d^{4}}, \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{96 \, b d^{4}}\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 \left (a + b x^{2}\right )^{\frac{5}{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 = 1.23526, size = 284, normalized size = 1.73 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (2 \,{\left (b x^{2} + a\right )}{\left (\frac{4 \,{\left (b x^{2} + a\right )}}{b d} - \frac{5 \,{\left (b c d^{3} - a d^{4}\right )}}{b d^{5}}\right )} + \frac{15 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}}{b d^{5}}\right )} + \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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^{3}}\right )} b}{48 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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