Optimal. Leaf size=137 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{5/2} \sqrt{d}} \]
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Rubi [A] time = 0.141449, 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, 89, 78, 63, 217, 206} \[ -\frac{a^2 \sqrt{c+d x^2}}{3 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x^2} (b c-a d)^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{5/2} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} a (3 b c-a d)+\frac{3}{2} b (b c-a d) x}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx,x,x^2\right )}{3 b^2 (b c-a d)}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac{2 a (3 b c-2 a d) \sqrt{c+d x^2}}{3 b^2 (b c-a d)^2 \sqrt{a+b x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{2 b^2}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac{2 a (3 b c-2 a d) \sqrt{c+d x^2}}{3 b^2 (b c-a d)^2 \sqrt{a+b x^2}}+\frac{\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 )}{b^3}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac{2 a (3 b c-2 a d) \sqrt{c+d x^2}}{3 b^2 (b c-a d)^2 \sqrt{a+b x^2}}+\frac{\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 )}{b^3}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{3 b^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac{2 a (3 b c-2 a d) \sqrt{c+d x^2}}{3 b^2 (b c-a d)^2 \sqrt{a+b x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{5/2} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.483143, size = 214, normalized size = 1.56 \[ \frac{\sqrt{c+d x^2} \left (\frac{\left (a+b x^2\right ) \left (3 b^2 c^2-a^2 d^2\right )}{d (b c-a d)^2}+\frac{a^2}{a d-b c}-\frac{3 \left (a+b x^2\right ) \left (\sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}}-\sqrt{d} \sqrt{a+b x^2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )\right )}{d \sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}}}\right )}{3 b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 651, normalized size = 4.8 \begin{align*}{\frac{1}{6\,{b}^{2} \left ( ad-bc \right ) ^{2} \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) } \left ( -8\,\sqrt{bd}{x}^{4}{a}^{2}b{d}^{2}+12\,{x}^{4}a{b}^{2}cd\sqrt{bd}+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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{a}^{2}b{d}^{2}-6\,\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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}a{b}^{2}cd+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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{b}^{3}{c}^{2}-6\,{x}^{2}{a}^{3}{d}^{2}\sqrt{bd}+2\,\sqrt{bd}{x}^{2}{a}^{2}bcd+12\,{x}^{2}a{b}^{2}{c}^{2}\sqrt{bd}+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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{a}^{3}{d}^{2}-6\,\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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{a}^{2}bcd+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 ) \sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }a{b}^{2}{c}^{2}-6\,{a}^{3}cd\sqrt{bd}+10\,\sqrt{bd}{a}^{2}b{c}^{2} \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\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 [B] time = 3.48415, size = 1473, normalized size = 10.75 \begin{align*} \left [\frac{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{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 (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \,{\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{12 \,{\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} +{\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 2 \,{\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{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 (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \,{\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{6 \,{\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} +{\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 2 \,{\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x^{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^{5}}{\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 [B] time = 1.35565, size = 443, normalized size = 3.23 \begin{align*} -\frac{\frac{3 \, \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 )}^{2}\right )}{\sqrt{b d}} - \frac{8 \,{\left (3 \, \sqrt{b d} a b^{4} c^{2} - 5 \, \sqrt{b d} a^{2} b^{3} c d + 2 \, \sqrt{b d} a^{3} b^{2} d^{2} - 6 \, \sqrt{b d}{\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 )}^{2} a b^{2} c + 3 \, \sqrt{b d}{\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 )}^{2} a^{2} b d + 3 \, \sqrt{b d}{\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 )}^{4} a\right )}}{{\left (b^{2} c - a b d -{\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 )}^{2}\right )}^{3}}}{6 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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