3.1960 \(\int \frac{(c x)^m}{\left (1+\frac{b}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0611903, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.28187, size = 42, normalized size = 0.95 \[ \frac{\left (c x\right )^{m} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b}{x^{2}}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0863817, size = 91, normalized size = 2.07 \[ \frac{x \sqrt{\frac{b+x^2}{b}} (c x)^m \left (\, _2F_1\left (\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{x^2}{b}\right )-\, _2F_1\left (\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{x^2}{b}\right )\right )}{(m+2) \sqrt{\frac{b}{x^2}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{ \left ( cx \right ) ^{m} \left ( 1+{\frac{b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(1+b/x^2)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{{\left (\frac{b}{x^{2}} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b/x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x)^m/(b/x^2 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{m} x^{2}}{{\left (x^{2} + b\right )} \sqrt{\frac{x^{2} + b}{x^{2}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b/x^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral((c*x)^m*x^2/((x^2 + b)*sqrt((x^2 + b)/x^2)), x)

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Sympy [A]  time = 15.085, size = 54, normalized size = 1.23 \[ - \frac{c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**m*x*x**m*gamma(-m/2 - 1/2)*hyper((3/2, -m/2 - 1/2), (-m/2 + 1/2,), b*exp_pol
ar(I*pi)/x**2)/(2*gamma(-m/2 + 1/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{{\left (\frac{b}{x^{2}} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b/x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x)^m/(b/x^2 + 1)^(3/2), x)