Optimal. Leaf size=171 \[ \frac{\left (-32 a c+35 b^2-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{768 c^{9/2}}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c} \]
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Rubi [A] time = 0.151818, antiderivative size = 171, 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 = {1357, 742, 779, 612, 621, 206} \[ \frac{\left (-32 a c+35 b^2-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{768 c^{9/2}}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 742
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^{11} \sqrt{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 \sqrt{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c}+\frac{\operatorname{Subst}\left (\int x \left (-2 a-\frac{7 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{15 c}\\ &=\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c}+\frac{\left (35 b^2-32 a c-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}-\frac{\left (b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{96 c^3}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c}+\frac{\left (35 b^2-32 a c-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{768 c^4}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c}+\frac{\left (35 b^2-32 a c-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\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 )}{384 c^4}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c}+\frac{\left (35 b^2-32 a c-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}+\frac{b \left (7 b^2-12 a c\right ) \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 )}{768 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.13708, size = 164, normalized size = 0.96 \[ \frac{-\frac{\left (32 a c-35 b^2+42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac{5 \left (12 a b c-7 b^3\right ) \left (2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )\right )}{256 c^{7/2}}+x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{x}^{11}\sqrt{c{x}^{6}+b{x}^{3}+a}\, 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 = 1.70713, size = 863, normalized size = 5.05 \begin{align*} \left [\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, 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 (384 \, c^{5} x^{12} + 48 \, b c^{4} x^{9} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{23040 \, c^{5}}, -\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, 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 (384 \, c^{5} x^{12} + 48 \, b c^{4} x^{9} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{11520 \, c^{5}}\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^{11} \sqrt{a + b x^{3} + c x^{6}}\, 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 \sqrt{c x^{6} + b x^{3} + a} x^{11}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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