Optimal. Leaf size=78 \[ -\frac {4 a c (d x)^m \left (a+\frac {b}{\sqrt {\frac {c}{x}}}\right )^{3/2} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac {3}{2},-2 m-1;\frac {5}{2};\frac {b}{a \sqrt {\frac {c}{x}}}+1\right )}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 78, 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} \[ -\frac {4 a c (d x)^m \left (a+\frac {b}{\sqrt {\frac {c}{x}}}\right )^{3/2} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac {3}{2},-2 m-1;\frac {5}{2};\frac {b}{a \sqrt {\frac {c}{x}}}+1\right )}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 67
Rule 341
Rule 343
Rule 369
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} (d x)^m \, dx &=\operatorname {Subst}\left (\int \sqrt {a+\frac {b \sqrt {x}}{\sqrt {c}}} (d x)^m \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\operatorname {Subst}\left (\left (x^{-m} (d x)^m\right ) \int \sqrt {a+\frac {b \sqrt {x}}{\sqrt {c}}} x^m \, 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 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 \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 \left (a+\frac {b}{\sqrt {\frac {c}{x}}}\right )^{3/2} \left (-\frac {b}{a \sqrt {\frac {c}{x}}}\right )^{-2 m} (d x)^m \, _2F_1\left (\frac {3}{2},-1-2 m;\frac {5}{2};1+\frac {b}{a \sqrt {\frac {c}{x}}}\right )}{3 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 85, normalized size = 1.09 \[ \frac {4 x (d x)^m \sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \, _2F_1\left (-\frac {1}{2},-2 m-\frac {5}{2};-2 m-\frac {3}{2};-\frac {a \sqrt {\frac {c}{x}}}{b}\right )}{(4 m+5) \sqrt {\frac {a \sqrt {\frac {c}{x}}}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\frac {b}{\sqrt {\frac {c}{x}}}}\, \left (d x \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {\frac {c}{x}}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {\frac {c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________