Optimal. Leaf size=256 \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]
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Rubi [A] time = 0.285943, antiderivative size = 256, 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, 740, 834, 806, 724, 206} \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 740
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-7 b^2+16 a c\right )-3 b c x}{x^4 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{3 a \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{1}{4} b \left (35 b^2-116 a c\right )-c \left (7 b^2-16 a c\right ) x}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^2 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{8} \left (-105 b^4+460 a b^2 c-256 a^2 c^2\right )-\frac{1}{4} b c \left (35 b^2-116 a c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{9 a^3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}-\frac{\left (5 b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )}{48 a^4}\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac{\left (5 b \left (7 b^2-12 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 )}{24 a^4}\\ &=\frac{2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt{a+b x^3+c x^6}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.182314, size = 223, normalized size = 0.87 \[ \frac{\frac{2 \sqrt{a} \left (2 a^2 x^3 \left (-86 b^2 c x^3-7 b^3+244 b c^2 x^6+128 c^3 x^9\right )+8 a^3 \left (b^2+7 b c x^3+16 c^2 x^6\right )-32 a^4 c+5 a b^2 x^6 \left (7 b^2-106 b c x^3-92 c^2 x^6\right )+105 b^4 x^9 \left (b+c x^3\right )\right )}{x^9 \sqrt{a+b x^3+c x^6}}-15 b \left (48 a^2 c^2-40 a b^2 c+7 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{144 a^{9/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{10}} \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 = 3.65787, size = 1550, normalized size = 6.05 \begin{align*} \text{result too large to display} \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{1}{x^{10} \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{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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