3.2994 \(\int \frac{(d x)^m}{\sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{4 a c (d x)^m \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac{1}{2},-2 m-1;\frac{3}{2};\frac{b}{a \sqrt{\frac{c}{x}}}+1\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.205763, antiderivative size = 76, 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 \[ -\frac{4 a c (d x)^m \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac{1}{2},-2 m-1;\frac{3}{2};\frac{b}{a \sqrt{\frac{c}{x}}}+1\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.0008, size = 83, normalized size = 1.09 \[ - \frac{4 a c \left (\frac{c}{x}\right )^{- m - \frac{3}{2}} \left (\frac{c}{x}\right )^{m + \frac{3}{2}} \left (d x\right )^{m} \left (- \frac{b}{a \sqrt{\frac{c}{x}}}\right )^{- 2 m} \sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}{{}_{2}F_{1}\left (\begin{matrix} - 2 m - 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b}{a \sqrt{\frac{c}{x}}}} \right )}}{b^{2}} \]

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)*sq
rt(a + b/sqrt(c/x))*hyper((-2*m - 1, 1/2), (3/2,), 1 + b/(a*sqrt(c/x)))/b**2

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Mathematica [A]  time = 0.57146, 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.

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{a+{b{\frac{1}{\sqrt{{\frac{c}{x}}}}}}}}}}\, dx \]

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. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{a + \frac{b}{\sqrt{\frac{c}{x}}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)