3.3045 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^m \, dx\)

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

[Out]

(Sqrt[a + b*Sqrt[d/x] + c/x]*x^(1 + m)*AppellF1[-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[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]))])

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Rubi [A]  time = 1.56698, 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 \[ \frac{x^{m+1} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} F_1\left (-2 (m+1);-\frac{1}{2},-\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*Sqrt[d/x] + c/x]*x^(1 + m)*AppellF1[-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[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]))])

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Rubi in Sympy [A]  time = 75.9263, size = 211, normalized size = 0.92 \[ \frac{d x^{m} \left (\frac{d}{x}\right )^{m} \left (\frac{d}{x}\right )^{- m - 1} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \operatorname{appellf_{1}}{\left (- 2 m - 2,- \frac{1}{2},- \frac{1}{2},- 2 m - 1,- \frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} - \sqrt{- 4 a c + b^{2} d}\right )},- \frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} + \sqrt{- 4 a c + b^{2} d}\right )} \right )}}{\left (m + 1\right ) \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} - \sqrt{- 4 a c + b^{2} d}\right )} + 1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d} + \sqrt{- 4 a c + b^{2} d}\right )} + 1}} \]

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

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Mathematica [A]  time = 0.114251, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^m \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{a+{\frac{c}{x}}+b\sqrt{{\frac{d}{x}}}}\, dx \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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