Optimal. Leaf size=225 \[ \frac{\left (8 a^2 c^2+12 a b c+5 b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^3 d^3}-\frac{b \left (8 a^2 c^2+12 a b c+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right )}{16 a^{7/2} d^3}-\frac{(8 a c+5 b) \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 a^2 d^3}+\frac{x^2 \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{6 a d^2} \]
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Rubi [A] time = 0.6221, antiderivative size = 267, normalized size of antiderivative = 1.19, 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} \[ \frac{\left (8 a^2 c^2+12 a b c+5 b^2\right ) \left (a \left (c+d x^2\right )+b\right )}{16 a^3 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{b \left (8 a^2 c^2+12 a b c+5 b^2\right ) \sqrt{a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{a \left (c+d x^2\right )+b}}\right )}{16 a^{7/2} d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(8 a c+5 b) \left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}} \]
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 \frac{x^5}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^5 \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^5 \sqrt{c+d x^2}}{\sqrt{b+a c+a d x^2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \operatorname{Subst}\left (\int \frac{x^2 \sqrt{c+d x}}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{b+a \left (c+d x^2\right )} \operatorname{Subst}\left (\int \frac{\sqrt{c+d x} \left (-c (b+a c)-\frac{1}{2} (5 b+8 a c) d x\right )}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{6 a d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (\left (-2 a c (b+a c) d^2+\frac{1}{2} (5 b+8 a c) d \left (\frac{a c d}{2}+\frac{3}{2} (b+a c) d\right )\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{12 a^2 d^4 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (5 b+8 a c) d \left (\frac{a c d}{2}+\frac{3}{2} (b+a c) d\right )\right ) \sqrt{b+a \left (c+d x^2\right )}\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^3 d^4 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (5 b+8 a c) d \left (\frac{a c d}{2}+\frac{3}{2} (b+a c) d\right )\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^2}} \, dx,x,\sqrt{c+d x^2}\right )}{12 a^3 d^5 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (-2 a c (b+a c) d^2+\frac{1}{2} (5 b+8 a c) d \left (\frac{a c d}{2}+\frac{3}{2} (b+a c) d\right )\right ) \sqrt{b+a \left (c+d x^2\right )}\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^3 d^5 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\left (5 b^2+12 a b c+8 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^3 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(5 b+8 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^2 \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{6 a d^2 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{b \left (5 b^2+12 a b c+8 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{16 a^{7/2} d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ \end{align*}
Mathematica [A] time = 0.269933, size = 140, normalized size = 0.62 \[ \frac{\sqrt{a} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (8 a^2 \left (c^2-c d x^2+d^2 x^4\right )+2 a b \left (13 c-5 d x^2\right )+15 b^2\right )-3 b \left (8 a^2 c^2+12 a b c+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{48 a^{7/2} d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 533, normalized size = 2.4 \begin{align*}{\frac{d{x}^{2}+c}{96\,{a}^{3}{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}}-36\,\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-36\,\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}}-15\,\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+30\,\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 = 1.45516, size = 961, normalized size = 4.27 \begin{align*} \left [\frac{3 \,{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, 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} - 10 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} + 26 \, a^{2} b c^{2} + 15 \, a b^{2} c +{\left (16 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{4} d^{3}}, \frac{3 \,{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, 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} - 10 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} + 26 \, a^{2} b c^{2} + 15 \, a b^{2} c +{\left (16 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{4} 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^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{a + \frac{b}{d x^{2} + c}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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