Optimal. Leaf size=141 \[ \frac {a x^4 \sqrt {a+b x^3+c x^6} F_1\left (\frac {4}{3};-\frac {3}{2},-\frac {3}{2};\frac {7}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.13, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1385, 510} \[ \frac {a x^4 \sqrt {a+b x^3+c x^6} F_1\left (\frac {4}{3};-\frac {3}{2},-\frac {3}{2};\frac {7}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1385
Rubi steps
\begin {align*} \int x^3 \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 x^3 \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^4 \sqrt {a+b x^3+c x^6} F_1\left (\frac {4}{3};-\frac {3}{2},-\frac {3}{2};\frac {7}{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 )}{4 \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*}
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Mathematica [B] time = 0.87, size = 453, normalized size = 3.21 \[ \frac {x \left (27 x^3 \left (640 a^2 c^2-404 a b^2 c+55 b^4\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )+8 \left (4 a^2 c \left (459 b+1280 c x^3\right )+a \left (-297 b^3+2052 b^2 c x^3+10204 b c^2 x^6+7360 c^3 x^9\right )-297 b^4 x^3-81 b^3 c x^6+3464 b^2 c^2 x^9+5488 b c^3 x^{12}+2240 c^4 x^{15}\right )+216 a b \left (11 b^2-68 a c\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )\right )}{232960 c^2 \sqrt {a+b x^3+c x^6}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{9} + b x^{6} + a x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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