Optimal. Leaf size=76 \[ -\frac {4 a c (d x)^m \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac {1}{2},-2 m-1;\frac {3}{2};\frac {b}{a \sqrt {\frac {c}{x}}}+1\right )}{b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {369, 343, 341, 67, 65} \[ -\frac {4 a c (d x)^m \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac {1}{2},-2 m-1;\frac {3}{2};\frac {b}{a \sqrt {\frac {c}{x}}}+1\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 341
Rule 343
Rule 369
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \, dx &=\operatorname {Subst}\left (\int \frac {(d x)^m}{\sqrt {a+\frac {b \sqrt {x}}{\sqrt {c}}}} \, 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 \sqrt {x}}{\sqrt {c}}}} \, 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 x}{\sqrt {c}}}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (\frac {\left (2 a \sqrt {c} \left (-\frac {b \sqrt {x}}{a \sqrt {c}}\right )^{-2 m} (d x)^m\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {b x}{a \sqrt {c}}\right )^{-1+2 (1+m)}}{\sqrt {a+\frac {b x}{\sqrt {c}}}} \, dx,x,\sqrt {x}\right )}{b},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {4 a c \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} (d x)^m \, _2F_1\left (\frac {1}{2},-1-2 m;\frac {3}{2};1+\frac {b}{a \sqrt {\frac {c}{x}}}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 116, normalized size = 1.53 \[ \frac {a^2 c (d x)^m \left (\frac {a \sqrt {\frac {c}{x}}}{a \sqrt {\frac {c}{x}}+b}\right )^{2 m-\frac {1}{2}} \, _2F_1\left (2 m+2,2 m+\frac {5}{2};2 m+3;\frac {b}{\sqrt {\frac {c}{x}} a+b}\right )}{(m+1) \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \left (a \sqrt {\frac {c}{x}}+b\right )^2} \]
Antiderivative was successfully verified.
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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 \[ \mathit {sage}_{0} x \]
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 \[ \int \frac {\left (d x \right )^{m}}{\sqrt {a +\frac {b}{\sqrt {\frac {c}{x}}}}}\, dx \]
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 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}}} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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