Optimal. Leaf size=195 \[ -\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^3 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}+\frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{8 c^{7/2}}+\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.228535, antiderivative size = 195, 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, 738, 832, 779, 621, 206} \[ -\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^3 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}+\frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{8 c^{7/2}}+\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 738
Rule 832
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{14}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2 (6 a+3 b x)}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3 \left (b^2-4 a c\right )}\\ &=\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-6 a b-\frac{3}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{9 c \left (b^2-4 a c\right )}\\ &=\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^3\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}+\frac{\left (5 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{8 c^3}\\ &=\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^3\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}+\frac{\left (5 b^2-4 a c\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 )}{4 c^3}\\ &=\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c \left (5 b^2-12 a c\right ) x^3\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}+\frac{\left (5 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 )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.191698, size = 181, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{c} \left (4 a^2 c \left (6 c x^3-13 b\right )+a \left (-62 b^2 c x^3+15 b^3-20 b c^2 x^6+8 c^3 x^9\right )+b^2 x^3 \left (15 b^2+5 b c x^3-2 c^2 x^6\right )\right )}{\sqrt{a+b x^3+c x^6}}-3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{24 c^{7/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{{x}^{14} \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 = 2.27616, size = 1272, normalized size = 6.52 \begin{align*} \left [-\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{6} + 5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{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 (2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{9} - 5 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{6} - 15 \, a b^{3} c + 52 \, a^{2} b c^{2} -{\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{48 \,{\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} +{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{6} +{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3}\right )}}, -\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{6} + 5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{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 (2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{9} - 5 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{6} - 15 \, a b^{3} c + 52 \, a^{2} b c^{2} -{\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{24 \,{\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} +{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{6} +{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3}\right )}}\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^{14}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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