3.1958 \(\int \sqrt{1+\frac{b}{x^2}} (c x)^m \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0569861, 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 \[ \frac{(c x)^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + b/x^2]*(c*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.30985, size = 44, normalized size = 1. \[ \frac{\left (c x\right )^{m} \left (\frac{1}{x}\right )^{m} \left (\frac{1}{x}\right )^{- m - 1}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{m}{2} - \frac{1}{2} \\ - \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b}{x^{2}}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0245206, size = 58, normalized size = 1.32 \[ \frac{x \sqrt{\frac{b}{x^2}+1} (c x)^m \, _2F_1\left (-\frac{1}{2},\frac{m}{2};\frac{m}{2}+1;-\frac{x^2}{b}\right )}{m \sqrt{\frac{b+x^2}{b}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \[ \int \sqrt{1+{\frac{b}{{x}^{2}}}} \left ( cx \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+b/x^2)^(1/2)*(c*x)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*sqrt(b/x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.90338, size = 48, normalized size = 1.09 \[ - \frac{\sqrt{b} c^{m} x^{m} \Gamma \left (- \frac{m}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} \\ \frac{m}{2} + 1 \end{matrix}\middle |{\frac{x^{2} e^{i \pi }}{b}} \right )}}{2 \Gamma \left (- \frac{m}{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*c**m*x**m*gamma(-m/2)*hyper((-1/2, m/2), (m/2 + 1,), x**2*exp_polar(I*p
i)/b)/(2*gamma(-m/2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*sqrt(b/x^2 + 1), x)