3.1962 \(\int \left (a+\frac{b}{x^2}\right )^p (c x)^m \, dx\)

Optimal. Leaf size=70 \[ \frac{(c x)^{m+1} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-m-1),-p;\frac{1-m}{2};-\frac{b}{a x^2}\right )}{c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0866844, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(c x)^{m+1} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-m-1),-p;\frac{1-m}{2};-\frac{b}{a x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^p*(c*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.658, size = 63, normalized size = 0.9 \[ \frac{\left (c x\right )^{m} \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (a + \frac{b}{x^{2}}\right )^{p} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b}{a x^{2}}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0628882, size = 73, normalized size = 1.04 \[ \frac{x (c x)^m \left (a+\frac{b}{x^2}\right )^p \left (\frac{a x^2}{b}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (m-2 p+1),-p;\frac{1}{2} (m-2 p+1)+1;-\frac{a x^2}{b}\right )}{m-2 p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( cx \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^p*(c*x)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*(a + b/x^2)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 106.656, size = 60, normalized size = 0.86 \[ - \frac{a^{p} c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**p*c**m*x*x**m*gamma(-m/2 - 1/2)*hyper((-p, -m/2 - 1/2), (-m/2 + 1/2,), b*exp
_polar(I*pi)/(a*x**2))/(2*gamma(-m/2 + 1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*(a + b/x^2)^p, x)