Optimal. Leaf size=136 \[ \frac{a x \sqrt{a+b x^3+c x^6} F_1\left (\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.0664819, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1348, 429} \[ \frac{a x \sqrt{a+b x^3+c x^6} F_1\left (\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 1348
Rule 429
Rubi steps
\begin{align*} \int \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a+b x^3+c x^6}\right ) \int \left (1+\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )^{3/2} \, dx}{\sqrt{1+\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}}}\\ &=\frac{a x \sqrt{a+b x^3+c x^6} F_1\left (\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{1+\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}}}\\ \end{align*}
Mathematica [B] time = 0.661665, size = 408, normalized size = 3. \[ \frac{x \left (8 \left (364 a^2 c+27 a b^2+548 a b c x^3+476 a c^2 x^6+211 b^2 c x^6+27 b^3 x^3+296 b c^2 x^9+112 c^3 x^{12}\right )-27 b x^3 \left (5 b^2-44 a c\right ) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-216 a \left (b^2-28 a c\right ) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )}{8960 c \sqrt{a+b x^3+c x^6}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}, x\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 \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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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