Optimal. Leaf size=216 \[ -\frac{\left (-8 a^2 c^2+4 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^2 d^3}+\frac{b \left (8 a^2 c^2+4 a b c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right )}{16 a^{5/2} d^3}+\frac{\left (c+d x^2\right )^3 \left (\frac{a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{6 a d^3}-\frac{(4 a c+b) \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 a d^3} \]
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Rubi [A] time = 0.620321, antiderivative size = 259, normalized size of antiderivative = 1.2, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6722, 1975, 446, 90, 80, 50, 63, 217, 206} \[ \frac{\left (8 a^2 c^2+4 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a^2 d^3}+\frac{b \left (8 a^2 c^2+4 a b c+b^2\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{a \left (c+d x^2\right )+b}}\right )}{16 a^{5/2} d^3 \sqrt{a \left (c+d x^2\right )+b}}-\frac{(8 a c+3 b) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{6 a d^2} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 446
Rule 90
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^5 \sqrt{a+\frac{b}{c+d x^2}} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^5 \sqrt{b+a \left (c+d x^2\right )}}{\sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^5 \sqrt{b+a c+a d x^2}}{\sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{b+a c+a d x}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x} \left (-c (b+a c)-\frac{1}{2} (3 b+8 a c) d x\right )}{\sqrt{c+d x}} \, dx,x,x^2\right )}{6 a d^2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{(3 b+8 a c) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{\left (\left (-2 a c (b+a c) d^2+\frac{1}{2} (3 b+8 a c) d \left (\frac{3 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{12 a^2 d^4 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (b^2+4 a b c+8 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a^2 d^3}-\frac{(3 b+8 a c) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (3 b+8 a c) d \left (\frac{3 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{24 a^2 d^4 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (b^2+4 a b c+8 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a^2 d^3}-\frac{(3 b+8 a c) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (3 b+8 a c) d \left (\frac{3 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^2}} \, dx,x,\sqrt{c+d x^2}\right )}{12 a^2 d^5 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (b^2+4 a b c+8 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a^2 d^3}-\frac{(3 b+8 a c) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (3 b+8 a c) d \left (\frac{3 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{12 a^2 d^5 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (b^2+4 a b c+8 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a^2 d^3}-\frac{(3 b+8 a c) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{6 a d^2}+\frac{b \left (b^2+4 a b c+8 a^2 c^2\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{16 a^{5/2} d^3 \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}
Mathematica [A] time = 0.309146, size = 137, normalized size = 0.63 \[ \frac{\sqrt{a} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (8 a^2 \left (c^2-c d x^2+d^2 x^4\right )+2 a b \left (d x^2-5 c\right )-3 b^2\right )+3 b \left (8 a^2 c^2+4 a b c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{48 a^{5/2} d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 533, normalized size = 2.5 \begin{align*}{\frac{d{x}^{2}+c}{96\,{a}^{2}{d}^{3}}\sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}} \left ( -48\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}{x}^{2}c{a}^{2}d\sqrt{a{d}^{2}}-12\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}{x}^{2}bad\sqrt{a{d}^{2}}+24\,\ln \left ( 1/2\,{\frac{2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd}{\sqrt{a{d}^{2}}}} \right ){a}^{2}b{c}^{2}d+12\,\ln \left ( 1/2\,{\frac{2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd}{\sqrt{a{d}^{2}}}} \right ){b}^{2}cad+16\, \left ( a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc \right ) ^{3/2}a\sqrt{a{d}^{2}}-36\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}cba\sqrt{a{d}^{2}}+3\,\ln \left ( 1/2\,{\frac{2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd}{\sqrt{a{d}^{2}}}} \right ){b}^{3}d-6\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}{b}^{2}\sqrt{a{d}^{2}} \right ){\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}{\frac{1}{\sqrt{a{d}^{2}}}}} \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 = 1.37123, size = 944, normalized size = 4.37 \begin{align*} \left [\frac{3 \,{\left (8 \, a^{2} b c^{2} + 4 \, a b^{2} c + b^{3}\right )} \sqrt{a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \,{\left (2 \, a d^{2} x^{4} +{\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt{a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \,{\left (8 \, a^{3} d^{3} x^{6} + 2 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 10 \, a^{2} b c^{2} - 3 \, a b^{2} c -{\left (8 \, a^{2} b c + 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{3} d^{3}}, -\frac{3 \,{\left (8 \, a^{2} b c^{2} + 4 \, a b^{2} c + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt{-a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \,{\left (8 \, a^{3} d^{3} x^{6} + 2 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 10 \, a^{2} b c^{2} - 3 \, a b^{2} c -{\left (8 \, a^{2} b c + 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{3} 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 x^{5} \sqrt{\frac{a c + a d x^{2} + b}{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.68737, size = 408, normalized size = 1.89 \begin{align*} \frac{1}{48} \, \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}{\left (2 \,{\left (\frac{4 \, x^{2} \mathrm{sgn}\left (d x^{2} + c\right )}{d} - \frac{4 \, a^{3} c d^{4} \mathrm{sgn}\left (d x^{2} + c\right ) - a^{2} b d^{4} \mathrm{sgn}\left (d x^{2} + c\right )}{a^{3} d^{6}}\right )} x^{2} + \frac{8 \, a^{3} c^{2} d^{3} \mathrm{sgn}\left (d x^{2} + c\right ) - 10 \, a^{2} b c d^{3} \mathrm{sgn}\left (d x^{2} + c\right ) - 3 \, a b^{2} d^{3} \mathrm{sgn}\left (d x^{2} + c\right )}{a^{3} d^{6}}\right )} - \frac{{\left (8 \, a^{3} b c^{2} d^{4} \mathrm{sgn}\left (d x^{2} + c\right ) + 4 \, a^{2} b^{2} c d^{4} \mathrm{sgn}\left (d x^{2} + c\right ) + a b^{3} d^{4} \mathrm{sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, a^{\frac{3}{2}} c d - 2 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a{\left | d \right |} - \sqrt{a} b d \right |}\right )}{32 \, a^{\frac{7}{2}} d^{6}{\left | d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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