Optimal. Leaf size=231 \[ \frac{\left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{11/2}}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c x^3 \left (21 b^2-20 a c\right )\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c} \]
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Rubi [A] time = 0.299663, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1357, 742, 832, 779, 612, 621, 206} \[ \frac{\left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{11/2}}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c x^3 \left (21 b^2-20 a c\right )\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 742
Rule 832
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^{14} \sqrt{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^4 \sqrt{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}+\frac{\operatorname{Subst}\left (\int x^2 \left (-3 a-\frac{9 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{18 c}\\ &=-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}+\frac{\operatorname{Subst}\left (\int x \left (9 a b+\frac{3}{4} \left (21 b^2-20 a c\right ) x\right ) \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{90 c^2}\\ &=-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}+\frac{\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{384 c^4}\\ &=\frac{\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{\left (\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3072 c^5}\\ &=\frac{\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{\left (\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\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 )}{1536 c^5}\\ &=\frac{\left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c \left (21 b^2-20 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{\left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.165049, size = 208, normalized size = 0.9 \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (16 b c^2 \left (113 a^2-34 a c x^6+8 c^2 x^{12}\right )+160 c^3 x^3 \left (-3 a^2+2 a c x^6+8 c^2 x^{12}\right )+16 b^2 c^2 x^3 \left (56 a-9 c x^6\right )+168 b^3 c \left (c x^6-10 a\right )-210 b^4 c x^3+315 b^5\right )-15 \left (240 a^2 b^2 c^2-64 a^3 c^3-140 a b^4 c+21 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{46080 c^{11/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{x}^{14}\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 = 1.75189, size = 1071, normalized size = 4.64 \begin{align*} \left [-\frac{15 \,{\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 128 \, b c^{5} x^{12} - 16 \,{\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} x^{9} + 8 \,{\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} x^{6} + 315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3} - 2 \,{\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{92160 \, c^{6}}, \frac{15 \,{\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{15} + 128 \, b c^{5} x^{12} - 16 \,{\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} x^{9} + 8 \,{\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} x^{6} + 315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3} - 2 \,{\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{46080 \, c^{6}}\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^{14} \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^{14}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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