Optimal. Leaf size=293 \[ \frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{16384 c^6}+\frac{\left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{32768 c^{13/2}}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c x^3 \left (33 b^2-28 a c\right )\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c} \]
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Rubi [A] time = 0.402334, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{16384 c^6}+\frac{\left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{32768 c^{13/2}}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c x^3 \left (33 b^2-28 a c\right )\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c} \]
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} \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^4 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}+\frac{\operatorname{Subst}\left (\int x^2 \left (-3 a-\frac{11 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{24 c}\\ &=-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}+\frac{\operatorname{Subst}\left (\int x \left (11 a b+\frac{3}{4} \left (33 b^2-28 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{168 c^2}\\ &=-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac{\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{768 c^4}\\ &=\frac{\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}-\frac{\left (\left (b^2-4 a c\right ) \left (33 b^4-72 a b^2 c+16 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^3\right )}{4096 c^5}\\ &=-\frac{\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac{\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac{\left (\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{32768 c^6}\\ &=-\frac{\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac{\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac{\left (\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{16384 c^6}\\ &=-\frac{\left (b^2-4 a c\right ) \left (33 b^4-72 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}}{16384 c^6}+\frac{\left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{6144 c^5}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{336 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{5/2}}{24 c}-\frac{\left (3 b \left (77 b^2-124 a c\right )-10 c \left (33 b^2-28 a c\right ) x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{13440 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (33 b^4-72 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 )}{32768 c^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.359141, size = 241, normalized size = 0.82 \[ \frac{\frac{\left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \left (2 \sqrt{c} \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6} \left (4 c \left (5 a+2 c x^6\right )-3 b^2+8 b c x^3\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )\right )}{4096 c^{11/2}}+\frac{\left (372 a b c-280 a c^2 x^3+330 b^2 c x^3-231 b^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{560 c^3}+x^9 \left (a+b x^3+c x^6\right )^{5/2}-\frac{11 b x^6 \left (a+b x^3+c x^6\right )^{5/2}}{14 c}}{24 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, 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 = 1.89127, size = 1570, normalized size = 5.36 \begin{align*} \left [\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{21} + 87040 \, b c^{7} x^{18} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{15} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{12} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{9} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{6} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{6881280 \, c^{7}}, -\frac{105 \,{\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{21} + 87040 \, b c^{7} x^{18} + 1280 \,{\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{15} - 128 \,{\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{12} + 16 \,{\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{9} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} - 8 \,{\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{6} + 2 \,{\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a}}{3440640 \, c^{7}}\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} \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{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{14}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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