3.2991 \(\int \sqrt{a+\frac{b}{\left (\frac{c}{x}\right )^{3/2}}} (d x)^m \, dx\)

Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{a+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{c^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}\right )}{(m+1) \sqrt{\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{a c^3}+1}} \]

[Out]

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

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Rubi [A]  time = 0.240272, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{x (d x)^m \sqrt{a+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{c^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}\right )}{(m+1) \sqrt{\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{a c^3}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.8125, size = 100, normalized size = 0.98 \[ \frac{c \left (\frac{c}{x}\right )^{m} \left (\frac{c}{x}\right )^{- m - \frac{3}{2}} \left (\frac{c}{x}\right )^{- m - 1} \left (\frac{c}{x}\right )^{m + \frac{3}{2}} \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (\frac{c}{x}\right )^{\frac{3}{2}}}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2 m}{3} + \frac{2}{3} \\ \frac{2 m}{3} + \frac{5}{3} \end{matrix}\middle |{- \frac{b}{a \left (\frac{c}{x}\right )^{\frac{3}{2}}}} \right )}}{\sqrt{1 + \frac{b}{a \left (\frac{c}{x}\right )^{\frac{3}{2}}}} \left (m + 1\right )} \]

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

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Mathematica [A]  time = 0.116193, size = 0, normalized size = 0. \[ \int \sqrt{a+\frac{b}{\left (\frac{c}{x}\right )^{3/2}}} (d x)^m \, dx \]

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 = 0.068, 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. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (\frac{c}{x}\right )^{\frac{3}{2}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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