Optimal. Leaf size=153 \[ \frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{7/2}}-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c} \]
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Rubi [A] time = 0.129359, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1114, 742, 640, 612, 621, 206} \[ \frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{7/2}}-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 742
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^5 \sqrt{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}+\frac{\operatorname{Subst}\left (\int \left (-a-\frac{5 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{8 c}\\ &=-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}+\frac{\left (5 b^2-4 a c\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{32 c^2}\\ &=\frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{256 c^3}\\ &=\frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{128 c^3}\\ &=\frac{\left (5 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^3}-\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0698704, size = 136, normalized size = 0.89 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left (b \left (8 c^2 x^4-52 a c\right )+24 c^2 x^2 \left (a+2 c x^4\right )-10 b^2 c x^2+15 b^3\right )-3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{768 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 247, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{8\,c} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b}{48\,{c}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{x}^{2}}{64\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{3}}{128\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{2}a}{32}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,{b}^{4}}{256}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{a{x}^{2}}{16\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{ab}{32\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{a}^{2}}{16}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{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 = 1.66946, size = 698, normalized size = 4.56 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{1536 \, c^{4}}, \frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \,{\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{768 \, c^{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 x^{5} \sqrt{a + b x^{2} + c x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43727, size = 197, normalized size = 1.29 \begin{align*} \frac{1}{384} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (6 \, x^{2} + \frac{b}{c}\right )} x^{2} - \frac{5 \, b^{2} c^{3} - 12 \, a c^{4}}{c^{5}}\right )} x^{2} + \frac{15 \, b^{3} c^{2} - 52 \, a b c^{3}}{c^{5}}\right )} + \frac{{\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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