Optimal. Leaf size=90 \[ \frac {(d x)^{m+1} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, _2F_1\left (-\frac {1}{2},\frac {1}{3} (-m-1);\frac {2-m}{3};-\frac {b}{a \left (c x^2\right )^{3/2}}\right )}{d (m+1) \sqrt {\frac {b}{a \left (c x^2\right )^{3/2}}+1}} \]
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Rubi [A] time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {368, 339, 365, 364} \[ \frac {(d x)^{m+1} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, _2F_1\left (-\frac {1}{2},\frac {1}{3} (-m-1);\frac {2-m}{3};-\frac {b}{a \left (c x^2\right )^{3/2}}\right )}{d (m+1) \sqrt {\frac {b}{a \left (c x^2\right )^{3/2}}+1}} \]
Antiderivative was successfully verified.
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Rule 339
Rule 364
Rule 365
Rule 368
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx &=\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int \sqrt {a+\frac {b}{x^3}} x^m \, dx,x,\sqrt {c x^2}\right )}{d}\\ &=-\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int x^{-2-m} \sqrt {a+b x^3} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{d}\\ &=-\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}}\right ) \operatorname {Subst}\left (\int x^{-2-m} \sqrt {1+\frac {b x^3}{a}} \, dx,x,\frac {1}{\sqrt {c x^2}}\right )}{d \sqrt {1+\frac {b}{a \left (c x^2\right )^{3/2}}}}\\ &=\frac {(d x)^{1+m} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, _2F_1\left (-\frac {1}{2},\frac {1}{3} (-1-m);\frac {2-m}{3};-\frac {b}{a \left (c x^2\right )^{3/2}}\right )}{d (1+m) \sqrt {1+\frac {b}{a \left (c x^2\right )^{3/2}}}}\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} \sqrt {\frac {a c^{2} x^{4} + \sqrt {c x^{2}} b}{c^{2} x^{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\frac {b}{\left (c \,x^{2}\right )^{\frac {3}{2}}}}\, \left (d x \right )^{m}\, dx \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{2}\right )^{\frac {3}{2}}}}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{{\left (c\,x^2\right )}^{3/2}}} \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{2}\right )^{\frac {3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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