Optimal. Leaf size=140 \[ \frac {x^4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {a+b x^3+c x^6}} \]
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Rubi [A] time = 0.13, antiderivative size = 140, 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} \[ \frac {x^4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {a+b x^3+c x^6}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1385
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b x^3+c x^6}} \, dx &=\frac {\left (\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}}}\right ) \int \frac {x^3}{\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}}}} \, dx}{\sqrt {a+b x^3+c x^6}}\\ &=\frac {x^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}}} 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}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {a+b x^3+c x^6}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 168, normalized size = 1.20 \[ \frac {x^4 \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )}{4 \sqrt {a+b x^3+c x^6}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\sqrt {c x^{6} + b x^{3} + a}}, x\right ) \]
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 \[ \int \frac {x^{3}}{\sqrt {c x^{6} + b x^{3} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {c \,x^{6}+b \,x^{3}+a}}\, dx \]
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 \[ \int \frac {x^{3}}{\sqrt {c x^{6} + b x^{3} + a}}\,{d x} \]
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 \[ \int \frac {x^3}{\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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