3.2988 \(\int \sqrt{a+b \left (\frac{c}{x}\right )^{3/2}} (d x)^m \, dx\)

Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3}{x^3 \left (\frac{c}{x}\right )^{3/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt{\frac{b c^3}{a x^3 \left (\frac{c}{x}\right )^{3/2}}+1}} \]

[Out]

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

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Rubi [A]  time = 0.262212, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3}{x^3 \left (\frac{c}{x}\right )^{3/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt{\frac{b c^3}{a x^3 \left (\frac{c}{x}\right )^{3/2}}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c/x)^(3/2)]*(d*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.9894, size = 83, normalized size = 0.81 \[ \frac{c \left (\frac{c}{x}\right )^{m} \left (\frac{c}{x}\right )^{- m - 1} \left (d x\right )^{m} \sqrt{a + b \left (\frac{c}{x}\right )^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{2 m}{3} - \frac{2}{3} \\ - \frac{2 m}{3} + \frac{1}{3} \end{matrix}\middle |{- \frac{b \left (\frac{c}{x}\right )^{\frac{3}{2}}}{a}} \right )}}{\sqrt{1 + \frac{b \left (\frac{c}{x}\right )^{\frac{3}{2}}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(a+b*(c/x)**(3/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.106825, size = 0, normalized size = 0. \[ \int \sqrt{a+b \left (\frac{c}{x}\right )^{3/2}} (d x)^m \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c/x)^(3/2)]*(d*x)^m,x]

[Out]

Integrate[Sqrt[a + b*(c/x)^(3/2)]*(d*x)^m, x]

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b \left ({\frac{c}{x}} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(a+b*(c/x)^(3/2))^(1/2),x)

[Out]

int((d*x)^m*(a+b*(c/x)^(3/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \left (\frac{c}{x}\right )^{\frac{3}{2}} + a} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*(c/x)^(3/2) + a)*(d*x)^m,x, algorithm="maxima")

[Out]

integrate(sqrt(b*(c/x)^(3/2) + 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*(c/x)^(3/2) + a)*(d*x)^m,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \left (\frac{c}{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(a+b*(c/x)**(3/2))**(1/2),x)

[Out]

Integral((d*x)**m*sqrt(a + b*(c/x)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \left (\frac{c}{x}\right )^{\frac{3}{2}} + a} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*(c/x)^(3/2) + a)*(d*x)^m,x, algorithm="giac")

[Out]

integrate(sqrt(b*(c/x)^(3/2) + a)*(d*x)^m, x)