3.2990 \(\int \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} (d x)^m \, dx\)

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]

(-4*a*c*(a + b/Sqrt[c/x])^(3/2)*(d*x)^m*Hypergeometric2F1[3/2, -1 - 2*m, 5/2, 1
+ b/(a*Sqrt[c/x])])/(3*b^2*(-(b/(a*Sqrt[c/x])))^(2*m))

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Rubi [A]  time = 0.205803, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{4 a c (d x)^m \left (a+\frac{b}{\sqrt{\frac{c}{x}}}\right )^{3/2} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac{3}{2},-2 m-1;\frac{5}{2};\frac{b}{a \sqrt{\frac{c}{x}}}+1\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/Sqrt[c/x]]*(d*x)^m,x]

[Out]

(-4*a*c*(a + b/Sqrt[c/x])^(3/2)*(d*x)^m*Hypergeometric2F1[3/2, -1 - 2*m, 5/2, 1
+ b/(a*Sqrt[c/x])])/(3*b^2*(-(b/(a*Sqrt[c/x])))^(2*m))

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Rubi in Sympy [A]  time = 17.9298, size = 85, normalized size = 1.09 \[ - \frac{4 a c \left (\frac{c}{x}\right )^{- m - \frac{3}{2}} \left (\frac{c}{x}\right )^{m + \frac{3}{2}} \left (d x\right )^{m} \left (- \frac{b}{a \sqrt{\frac{c}{x}}}\right )^{- 2 m} \left (a + \frac{b}{\sqrt{\frac{c}{x}}}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - 2 m - 1, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b}{a \sqrt{\frac{c}{x}}}} \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(a+b/(c/x)**(1/2))**(1/2),x)

[Out]

-4*a*c*(c/x)**(-m - 3/2)*(c/x)**(m + 3/2)*(d*x)**m*(-b/(a*sqrt(c/x)))**(-2*m)*(a
+ b/sqrt(c/x))**(3/2)*hyper((-2*m - 1, 3/2), (5/2,), 1 + b/(a*sqrt(c/x)))/(3*b*
*2)

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Mathematica [A]  time = 0.0985244, 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]  Integrate[Sqrt[a + b/Sqrt[c/x]]*(d*x)^m,x]

[Out]

(4*Sqrt[a + b/Sqrt[c/x]]*x*(d*x)^m*Hypergeometric2F1[-1/2, -5/2 - 2*m, -3/2 - 2*
m, -((a*Sqrt[c/x])/b)])/((5 + 4*m)*Sqrt[1 + (a*Sqrt[c/x])/b])

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{{\frac{c}{x}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(a+b/(c/x)^(1/2))^(1/2),x)

[Out]

int((d*x)^m*(a+b/(c/x)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m*sqrt(a + b/sqrt(c/x)),x, algorithm="maxima")

[Out]

integrate((d*x)^m*sqrt(a + b/sqrt(c/x)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m*sqrt(a + b/sqrt(c/x)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(a+b/(c/x)**(1/2))**(1/2),x)

[Out]

Integral((d*x)**m*sqrt(a + b/sqrt(c/x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m*sqrt(a + b/sqrt(c/x)),x, algorithm="giac")

[Out]

integrate((d*x)^m*sqrt(a + b/sqrt(c/x)), x)