3.1961 \(\int \left (1+\frac{b}{x^2}\right )^p (c x)^m \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0531316, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{1}{2} (-m-1),-p;\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(1 + b/x^2)^p*(c*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.77472, size = 42, normalized size = 0.95 \[ \frac{\left (c x\right )^{m} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b}{x^{2}}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0555734, size = 71, normalized size = 1.61 \[ \frac{x \left (\frac{b}{x^2}+1\right )^p \left (\frac{x^2}{b}+1\right )^{-p} (c x)^m \, _2F_1\left (\frac{1}{2} (m-2 p+1),-p;\frac{1}{2} (m-2 p+1)+1;-\frac{x^2}{b}\right )}{m-2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.152, size = 0, normalized size = 0. \[ \int \left ( 1+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( cx \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+b/x^2)^p*(c*x)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*(b/x^2 + 1)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 83.708, size = 54, normalized size = 1.23 \[ - \frac{c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**m*x*x**m*gamma(-m/2 - 1/2)*hyper((-p, -m/2 - 1/2), (-m/2 + 1/2,), b*exp_pola
r(I*pi)/x**2)/(2*gamma(-m/2 + 1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*(b/x^2 + 1)^p, x)