Optimal. Leaf size=102 \[ \frac {x (d x)^m \sqrt {\frac {b c^3}{a x^3 \left (\frac {c}{x}\right )^{3/2}}+1} \, _2F_1\left (\frac {1}{2},-\frac {2}{3} (m+1);\frac {1}{3} (1-2 m);-\frac {b c^3}{a \left (\frac {c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt {a+\frac {b c^3}{x^3 \left (\frac {c}{x}\right )^{3/2}}}} \]
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Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {369, 343, 341, 339, 365, 364} \[ \frac {x (d x)^m \sqrt {\frac {b c^3}{a x^3 \left (\frac {c}{x}\right )^{3/2}}+1} \, _2F_1\left (\frac {1}{2},-\frac {2}{3} (m+1);\frac {1}{3} (1-2 m);-\frac {b c^3}{a \left (\frac {c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt {a+\frac {b c^3}{x^3 \left (\frac {c}{x}\right )^{3/2}}}} \]
Antiderivative was successfully verified.
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Rule 339
Rule 341
Rule 343
Rule 364
Rule 365
Rule 369
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\sqrt {a+b \left (\frac {c}{x}\right )^{3/2}}} \, dx &=\operatorname {Subst}\left (\int \frac {(d x)^m}{\sqrt {a+\frac {b c^{3/2}}{x^{3/2}}}} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\operatorname {Subst}\left (\left (x^{-m} (d x)^m\right ) \int \frac {x^m}{\sqrt {a+\frac {b c^{3/2}}{x^{3/2}}}} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\operatorname {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 (1+m)}}{\sqrt {a+\frac {b c^{3/2}}{x^3}}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1-2 (1+m)}}{\sqrt {a+b c^{3/2} x^3}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (\frac {\left (2 \sqrt {1+\frac {b c^{3/2}}{a x^{3/2}}} x^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1-2 (1+m)}}{\sqrt {1+\frac {b c^{3/2} x^3}{a}}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{\sqrt {a+\frac {b c^{3/2}}{x^{3/2}}}},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {\sqrt {1+\frac {b c^3}{a \left (\frac {c}{x}\right )^{3/2} x^3}} x (d x)^m \, _2F_1\left (\frac {1}{2},-\frac {2}{3} (1+m);\frac {1}{3} (1-2 m);-\frac {b c^3}{a \left (\frac {c}{x}\right )^{3/2} x^3}\right )}{(1+m) \sqrt {a+\frac {b c^3}{\left (\frac {c}{x}\right )^{3/2} x^3}}}\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\sqrt {a+b \left (\frac {c}{x}\right )^{3/2}}} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {b \left (\frac {c}{x}\right )^{\frac {3}{2}} + a}}\,{d x} \]
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 \[ \int \frac {\left (d x \right )^{m}}{\sqrt {a +\left (\frac {c}{x}\right )^{\frac {3}{2}} b}}\, dx \]
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 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {b \left (\frac {c}{x}\right )^{\frac {3}{2}} + a}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{\sqrt {a+b\,{\left (\frac {c}{x}\right )}^{3/2}}} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {a + b \left (\frac {c}{x}\right )^{\frac {3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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