Optimal. Leaf size=73 \[ \frac {\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (3,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a^3 d (m+1) \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1355, 364} \[ \frac {\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (3,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a^3 d (m+1) \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1355
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {(d x)^m}{\left (a b+b^2 x^3\right )^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {(d x)^{1+m} \left (a+b x^3\right ) \, _2F_1\left (3,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{a^3 d (1+m) \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.82 \[ \frac {x \left (a+b x^3\right ) (d x)^m \, _2F_1\left (3,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{a^3 (m+1) \sqrt {\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} \left (d x\right )^{m}}{b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}}, x\right ) \]
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 \[ \int \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (d x \right )^{m}}{\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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