Optimal. Leaf size=158 \[ \frac {a (d x)^{m+1} \sqrt {a+b x^3+c x^6} F_1\left (\frac {m+1}{3};-\frac {3}{2},-\frac {3}{2};\frac {m+4}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.15, antiderivative size = 158, 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} \[ \frac {a (d x)^{m+1} \sqrt {a+b x^3+c x^6} F_1\left (\frac {m+1}{3};-\frac {3}{2},-\frac {3}{2};\frac {m+4}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1385
Rubi steps
\begin {align*} \int (d x)^m \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 (d x)^m \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 (d x)^{1+m} \sqrt {a+b x^3+c x^6} F_1\left (\frac {1+m}{3};-\frac {3}{2},-\frac {3}{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 [B] time = 0.27, size = 357, normalized size = 2.26 \[ \frac {x (d x)^m \sqrt {a+b x^3+c x^6} \left (a \left (m^2+11 m+28\right ) F_1\left (\frac {m+1}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+4}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )+(m+1) x^3 \left (c (m+4) x^3 F_1\left (\frac {m+7}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+10}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )+b (m+7) F_1\left (\frac {m+4}{3};-\frac {1}{2},-\frac {1}{2};\frac {m+7}{3};-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{\sqrt {b^2-4 a c}-b}\right )\right )\right )}{(m+1) (m+4) (m+7) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}+b}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}} \left (d x \right )^{m}\, dx \]
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 \[ \int {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \]
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 \[ \int \left (d x\right )^{m} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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