3.3053 \(\int \frac{x^m}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=230 \[ \frac{x^{m+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1} F_1\left (-2 (m+1);\frac{1}{2},\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.32099, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^{m+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1} F_1\left (-2 (m+1);\frac{1}{2},\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 78.1134, size = 207, normalized size = 0.9 \[ \frac{d x^{m} \left (\frac{d}{x}\right )^{m} \left (\frac{d}{x}\right )^{- m - 1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} - \sqrt{- 4 a c + b^{2} d}\right )} + 1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} + \sqrt{- 4 a c + b^{2} d}\right )} + 1} \operatorname{appellf_{1}}{\left (- 2 m - 2,\frac{1}{2},\frac{1}{2},- 2 m - 1,- \frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} - \sqrt{- 4 a c + b^{2} d}\right )},- \frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} + \sqrt{- 4 a c + b^{2} d}\right )} \right )}}{\left (m + 1\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x**m*(d/x)**m*(d/x)**(-m - 1)*sqrt(2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d) - sqrt(-4
*a*c + b**2*d))) + 1)*sqrt(2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d) + sqrt(-4*a*c + b**
2*d))) + 1)*appellf1(-2*m - 2, 1/2, 1/2, -2*m - 1, -2*c*sqrt(d/x)/(sqrt(d)*(b*sq
rt(d) - sqrt(-4*a*c + b**2*d))), -2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d) + sqrt(-4*a*
c + b**2*d))))/((m + 1)*sqrt(a + b*sqrt(d/x) + c/x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.249975, size = 0, normalized size = 0. \[ \int \frac{x^m}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.017, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{a+{\frac{c}{x}}+b\sqrt{{\frac{d}{x}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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