Optimal. Leaf size=199 \[ -\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}+\frac{b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{768 a^{9/2}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}} \]
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Rubi [A] time = 0.226768, antiderivative size = 199, 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, 744, 834, 806, 720, 724, 206} \[ -\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}+\frac{b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{768 a^{9/2}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 744
Rule 834
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3+c x^6}}{x^{16}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^6} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{7 b}{2}+2 c x\right ) \sqrt{a+b x+c x^2}}{x^5} \, dx,x,x^3\right )}{15 a}\\ &=-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{4} \left (35 b^2-32 a c\right )+\frac{7 b c x}{2}\right ) \sqrt{a+b x+c x^2}}{x^4} \, dx,x,x^3\right )}{60 a^2}\\ &=-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}-\frac{\left (b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{96 a^3}\\ &=\frac{b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}+\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}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{768 a^4}\\ &=\frac{b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}-\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 a-x^2} \, dx,x,\frac{2 a+b x^3}{\sqrt{a+b x^3+c x^6}}\right )}{384 a^4}\\ &=\frac{b \left (7 b^2-12 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{15 a x^{15}}+\frac{7 b \left (a+b x^3+c x^6\right )^{3/2}}{120 a^2 x^{12}}-\frac{\left (35 b^2-32 a c\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 a^3 x^9}-\frac{b \left (7 b^2-12 a c\right ) \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 )}{768 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.170417, size = 173, normalized size = 0.87 \[ -\frac{\sqrt{a+b x^3+c x^6} \left (-8 a^2 x^6 \left (7 b^2+29 b c x^3+32 c^2 x^6\right )+16 a^3 \left (3 b x^3+8 c x^6\right )+384 a^4+10 a b^2 x^9 \left (7 b+46 c x^3\right )-105 b^4 x^{12}\right )}{5760 a^4 x^{15}}-\frac{b \left (48 a^2 c^2-40 a b^2 c+7 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{768 a^{9/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{16}}\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 = 2.6458, size = 910, normalized size = 4.57 \begin{align*} \left [\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt{a} x^{15} \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^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{12} - 2 \,{\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{9} - 48 \, a^{4} b x^{3} + 8 \,{\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{6} - 384 \, a^{5}\right )} \sqrt{c x^{6} + b x^{3} + a}}{23040 \, a^{5} x^{15}}, \frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt{-a} x^{15} \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^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{12} - 2 \,{\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{9} - 48 \, a^{4} b x^{3} + 8 \,{\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{6} - 384 \, a^{5}\right )} \sqrt{c x^{6} + b x^{3} + a}}{11520 \, a^{5} x^{15}}\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{\sqrt{a + b x^{3} + c x^{6}}}{x^{16}}\, 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{\sqrt{c x^{6} + b x^{3} + a}}{x^{16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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