Optimal. Leaf size=216 \[ -\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3072 a^{9/2}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}} \]
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Rubi [A] time = 0.207242, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 744, 806, 720, 724, 206} \[ -\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3072 a^{9/2}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 744
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{7 b}{2}+c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^3\right )}{18 a}\\ &=-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}+\frac{\left (7 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right )}{72 a^2}\\ &=-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{384 a^3}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3072 a^4}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^3}{\sqrt{a+b x^3+c x^6}}\right )}{1536 a^4}\\ &=\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3072 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.212798, size = 206, normalized size = 0.95 \[ -\frac{\frac{\left (\frac{7 b^2}{2}-2 a c\right ) \left (16 a^{3/2} \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}-3 x^6 \left (b^2-4 a c\right ) \left (2 \sqrt{a} \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}-x^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )\right )\right )}{256 a^{7/2} x^{12}}-\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{10 a x^{15}}+\frac{\left (a+b x^3+c x^6\right )^{5/2}}{x^{18}}}{18 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{19}} \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 = 3.90839, size = 1111, normalized size = 5.14 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{a} x^{18} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \,{\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{15} - 2 \,{\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{12} + 8 \,{\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{9} - 1664 \, a^{5} b x^{3} - 16 \,{\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{6} - 1280 \, a^{6}\right )} \sqrt{c x^{6} + b x^{3} + a}}{92160 \, a^{5} x^{18}}, \frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-a} x^{18} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \,{\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{15} - 2 \,{\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{12} + 8 \,{\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{9} - 1664 \, a^{5} b x^{3} - 16 \,{\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{6} - 1280 \, a^{6}\right )} \sqrt{c x^{6} + b x^{3} + a}}{46080 \, a^{5} x^{18}}\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{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{19}}\, 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{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{19}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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