Optimal. Leaf size=78 \[ \frac {x (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}} \, _2F_1\left (-\frac {1}{2},-\frac {2}{3} (m+1);\frac {1}{3} (1-2 m);-\frac {b}{a (c x)^{3/2}}\right )}{(m+1) \sqrt {\frac {b}{a (c x)^{3/2}}+1}} \]
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Rubi [A] time = 0.11, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {367, 343, 341, 339, 365, 364} \[ \frac {x (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}} \, _2F_1\left (-\frac {1}{2},-\frac {2}{3} (m+1);\frac {1}{3} (1-2 m);-\frac {b}{a (c x)^{3/2}}\right )}{(m+1) \sqrt {\frac {b}{a (c x)^{3/2}}+1}} \]
Antiderivative was successfully verified.
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Rule 339
Rule 341
Rule 343
Rule 364
Rule 365
Rule 367
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+\frac {b}{x^{3/2}}} \left (\frac {d x}{c}\right )^m \, dx,x,c x\right )}{c}\\ &=\frac {\left ((c x)^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int \sqrt {a+\frac {b}{x^{3/2}}} x^m \, dx,x,c x\right )}{c}\\ &=\frac {\left (2 (c x)^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int \sqrt {a+\frac {b}{x^3}} x^{-1+2 (1+m)} \, dx,x,\sqrt {c x}\right )}{c}\\ &=-\frac {\left (2 (c x)^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {a+b x^3} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{c}\\ &=-\frac {\left (2 (c x)^{-m} (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}}\right ) \operatorname {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {1+\frac {b x^3}{a}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{c \sqrt {1+\frac {b}{a (c x)^{3/2}}}}\\ &=\frac {x (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}} \, _2F_1\left (-\frac {1}{2},-\frac {2}{3} (1+m);\frac {1}{3} (1-2 m);-\frac {b}{a (c x)^{3/2}}\right )}{(1+m) \sqrt {1+\frac {b}{a (c x)^{3/2}}}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 84, normalized size = 1.08 \[ \frac {4 x (d x)^m \sqrt {a+\frac {b}{(c x)^{3/2}}} \, _2F_1\left (-\frac {1}{2},\frac {1}{6} (4 m+1);\frac {1}{6} (4 m+7);-\frac {a (c x)^{3/2}}{b}\right )}{(4 m+1) \sqrt {\frac {a (c x)^{3/2}+b}{b}}} \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x\right )^{\frac {3}{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\frac {b}{\left (c x \right )^{\frac {3}{2}}}}\, \left (d x \right )^{m}\, dx \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x\right )^{\frac {3}{2}}}}\,{d x} \]
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 \[ \int \sqrt {a+\frac {b}{{\left (c\,x\right )}^{3/2}}}\,{\left (d\,x\right )}^m \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x\right )^{\frac {3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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