Optimal. Leaf size=129 \[ -\frac{a^2 \sqrt{c+d x^2}}{b^2 \sqrt{a+b x^2} (b c-a d)}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d} \]
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Rubi [A] time = 0.164462, antiderivative size = 129, 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, 80, 63, 217, 206} \[ -\frac{a^2 \sqrt{c+d x^2}}{b^2 \sqrt{a+b x^2} (b c-a d)}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 \sqrt{c+d x^2}}{b^2 (b c-a d) \sqrt{a+b x^2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} a (b c-a d)+\frac{1}{2} b (b c-a d) x}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{b^2 (b c-a d)}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{b^2 (b c-a d) \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d}-\frac{(b c+3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{4 b^2 d}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{b^2 (b c-a d) \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d}-\frac{(b c+3 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 )}{2 b^3 d}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{b^2 (b c-a d) \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d}-\frac{(b c+3 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 )}{2 b^3 d}\\ &=-\frac{a^2 \sqrt{c+d x^2}}{b^2 (b c-a d) \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b^2 d}-\frac{(b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{5/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.345041, size = 185, normalized size = 1.43 \[ \frac{\sqrt{a+b x^2} \sqrt{b c-a d} \left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \sqrt{\frac{b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )-b \sqrt{d} \left (c+d x^2\right ) \left (-3 a^2 d+a b \left (c-d x^2\right )+b^2 c x^2\right )}{2 b^3 d^{3/2} \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 553, normalized size = 4.3 \begin{align*} -{\frac{1}{4\,{b}^{2}d \left ( ad-bc \right ) } \left ( 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 ){x}^{2}{a}^{2}b{d}^{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 ){x}^{2}a{b}^{2}cd-\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 ){x}^{2}{b}^{3}{c}^{2}-2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}abd+2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{x}^{2}{b}^{2}c+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}{d}^{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 ){a}^{2}bcd-\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{b}^{2}{c}^{2}-6\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }{a}^{2}d+2\,\sqrt{bd}\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }abc \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}}} \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 = 2.69498, size = 1049, normalized size = 8.13 \begin{align*} \left [\frac{{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} 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 (a b^{2} c d - 3 \, a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} +{\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2}\right )}}, \frac{{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} 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 (a b^{2} c d - 3 \, a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} +{\left (b^{5} c d^{2} - a 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{3}{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 = 1.24991, size = 254, normalized size = 1.97 \begin{align*} -\frac{\frac{8 \, \sqrt{b d} a^{2}}{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}} - \frac{2 \, \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}}{b d} - \frac{{\left (\sqrt{b d} b c + 3 \, \sqrt{b d} a d\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 )}^{2}\right )}{b d^{2}}}{4 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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