3.2989 \(\int \sqrt{a+b \sqrt{\frac{c}{x}}} (d x)^m \, dx\)

Optimal. Leaf size=60 \[ \frac{4 x^{m+1} \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \, _2F_1\left (1,\frac{1}{2} (-4 m-1);\frac{5}{2};\frac{a+b \sqrt{\frac{c}{x}}}{a}\right )}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.221122, antiderivative size = 80, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{4 b^2 c (d x)^m \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \, _2F_1\left (\frac{3}{2},2 m+3;\frac{5}{2};\frac{\sqrt{\frac{c}{x}} b}{a}+1\right )}{3 a^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*b^2*c*(a + b*Sqrt[c/x])^(3/2)*(-((b*Sqrt[c/x])/a))^(2*m)*(d*x)^m*Hypergeometr
ic2F1[3/2, 3 + 2*m, 5/2, 1 + (b*Sqrt[c/x])/a])/(3*a^3)

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Rubi in Sympy [A]  time = 15.8198, size = 65, normalized size = 1.08 \[ \frac{4 b^{2} c \left (d x\right )^{m} \left (- \frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} 2 m + 3, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b \sqrt{\frac{c}{x}}}{a}} \right )}}{3 a^{3}} \]

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

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Mathematica [A]  time = 0.068796, size = 78, normalized size = 1.3 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{\frac{c}{x}}} \, _2F_1\left (-\frac{1}{2},-2 (m+1);-2 m-1;-\frac{b \sqrt{\frac{c}{x}}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{\frac{c}{x}}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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