Optimal. Leaf size=157 \[ \frac {(d x)^{m+1} \sqrt {a+b x^3+c x^6} F_1\left (\frac {m+1}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+4}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.14, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1385, 510} \[ \frac {(d x)^{m+1} \sqrt {a+b x^3+c x^6} F_1\left (\frac {m+1}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+4}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1385
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a+b x^3+c x^6} \, dx &=\frac {\sqrt {a+b x^3+c x^6} \int (d x)^m \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 {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 {(d x)^{1+m} \sqrt {a+b x^3+c x^6} F_1\left (\frac {1+m}{3};-\frac {1}{2},-\frac {1}{2};\frac {4+m}{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 )}{d (1+m) \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 [A] time = 0.09, size = 181, normalized size = 1.15 \[ \frac {x (d x)^m \sqrt {a+b x^3+c x^6} F_1\left (\frac {m+1}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+4}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )}{(m+1) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}+b}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{6} + b x^{3} + a} \left (d x\right )^{m}, x\right ) \]
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 \[ \int \sqrt {c x^{6} + b x^{3} + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \sqrt {c \,x^{6}+b \,x^{3}+a}\, \left (d x \right )^{m}\, dx \]
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 \[ \int \sqrt {c x^{6} + b x^{3} + a} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,\sqrt {c\,x^6+b\,x^3+a} \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + b x^{3} + c x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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