3.1957 \(\int \left (1+\frac{b}{x^2}\right )^{3/2} (c x)^m \, dx\)

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 = 0.0704241, 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[(1 + b/x^2)^(3/2)*(c*x)^m,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 = 6.58781, size = 44, normalized size = 1. \[ \frac{\left (c x\right )^{m} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b}{x^{2}}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+b/x**2)**(3/2)*(c*x)**m,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 = 0.111911, size = 100, normalized size = 2.27 \[ \frac{\sqrt{\frac{b}{x^2}+1} (c x)^m \left ((m-2) x^2 \, _2F_1\left (-\frac{1}{2},\frac{m}{2};\frac{m}{2}+1;-\frac{x^2}{b}\right )+b m \, _2F_1\left (-\frac{1}{2},\frac{m}{2}-1;\frac{m}{2};-\frac{x^2}{b}\right )\right )}{(m-2) m x \sqrt{\frac{b+x^2}{b}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int \left ( 1+{\frac{b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( cx \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+b/x^2)^(3/2)*(c*x)^m,x)

[Out]

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

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

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. \[{\rm integral}\left (\frac{{\left (x^{2} + b\right )} \left (c x\right )^{m} \sqrt{\frac{x^{2} + b}{x^{2}}}}{x^{2}}, x\right ) \]

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

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Sympy [A]  time = 76.2196, size = 56, normalized size = 1.27 \[ - \frac{c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+b/x**2)**(3/2)*(c*x)**m,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_po
lar(I*pi)/x**2)/(2*gamma(-m/2 + 1/2))

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

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)