Optimal. Leaf size=150 \[ \frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{256 c^{7/2}}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
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Rubi [A] time = 0.116251, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 640, 612, 621, 206} \[ \frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{256 c^{7/2}}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^5 \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac{b \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{32 c^2}\\ &=\frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac{\left (b \left (b^2-4 a c\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{256 c^3}\\ &=\frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac{\left (b \left (b^2-4 a c\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^3}{\sqrt{a+b x^3+c x^6}}\right )}{128 c^3}\\ &=\frac{b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{256 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.152161, size = 149, normalized size = 0.99 \[ -\frac{b \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )-2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}\right )}{256 c^{7/2}}-\frac{b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \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 = 1.65705, size = 840, normalized size = 5.6 \begin{align*} \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (128 \, c^{5} x^{12} + 176 \, b c^{4} x^{9} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{7680 \, c^{4}}, \frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \,{\left (128 \, c^{5} x^{12} + 176 \, b c^{4} x^{9} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{3840 \, 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} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15868, size = 232, normalized size = 1.55 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x^{3} + 11 \, b\right )} x^{3} + \frac{b^{2} c^{3} + 32 \, a c^{4}}{c^{4}}\right )} x^{3} - \frac{5 \, b^{3} c^{2} - 28 \, a b c^{3}}{c^{4}}\right )} x^{3} + \frac{15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}}{c^{4}}\right )} + \frac{{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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