3.2993 \(\int \frac{(d x)^m}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

Optimal. Leaf size=58 \[ \frac{4 x^{m+1} \sqrt{a+b \sqrt{\frac{c}{x}}} \, _2F_1\left (1,\frac{1}{2} (-4 m-3);\frac{3}{2};\frac{a+b \sqrt{\frac{c}{x}}}{a}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.220988, antiderivative size = 78, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{4 b^2 c (d x)^m \sqrt{a+b \sqrt{\frac{c}{x}}} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \, _2F_1\left (\frac{1}{2},2 m+3;\frac{3}{2};\frac{\sqrt{\frac{c}{x}} b}{a}+1\right )}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*b^2*c*Sqrt[a + b*Sqrt[c/x]]*(-((b*Sqrt[c/x])/a))^(2*m)*(d*x)^m*Hypergeometric
2F1[1/2, 3 + 2*m, 3/2, 1 + (b*Sqrt[c/x])/a])/a^3

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Rubi in Sympy [A]  time = 16.0678, size = 63, normalized size = 1.09 \[ \frac{4 b^{2} c \left (d x\right )^{m} \left (- \frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \sqrt{a + b \sqrt{\frac{c}{x}}}{{}_{2}F_{1}\left (\begin{matrix} 2 m + 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b \sqrt{\frac{c}{x}}}{a}} \right )}}{a^{3}} \]

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

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Mathematica [A]  time = 0.288671, size = 96, normalized size = 1.66 \[ \frac{4 b^2 c (d x)^m \left (1-\frac{a}{a+b \sqrt{\frac{c}{x}}}\right )^{2 m} \, _2F_1\left (2 m+\frac{5}{2},2 m+3;2 m+\frac{7}{2};\frac{a}{a+b \sqrt{\frac{c}{x}}}\right )}{(4 m+5) \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*b^2*c*(1 - a/(a + b*Sqrt[c/x]))^(2*m)*(d*x)^m*Hypergeometric2F1[5/2 + 2*m, 3
+ 2*m, 7/2 + 2*m, a/(a + b*Sqrt[c/x])])/((5 + 4*m)*(a + b*Sqrt[c/x])^(5/2))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{a+b\sqrt{{\frac{c}{x}}}}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

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(-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(b*sqrt(c/x) + a),x, algorithm="giac")

[Out]

Exception raised: TypeError