3.433 \(\int (g x)^m \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=66 \[ \frac{(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{g (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0479034, antiderivative size = 66, 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 \[ \frac{(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right )}{g (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(g*x)^m*(a + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.98148, size = 51, normalized size = 0.77 \[ \frac{\left (g x\right )^{m + 1} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{g \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(g*x)**(m + 1)*(1 + c*x**2/a)**(-p)*(a + c*x**2)**p*hyper((-p, m/2 + 1/2), (m/2
+ 3/2,), -c*x**2/a)/(g*(m + 1))

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Mathematica [A]  time = 0.0356605, size = 64, normalized size = 0.97 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+1}{2}+1;-\frac{c x^2}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(c*x^2+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + a)^p*(g*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + a)^p*(g*x)^m, x)

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Sympy [A]  time = 107.458, size = 54, normalized size = 0.82 \[ \frac{a^{p} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**p*g**m*x*x**m*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), c*x**2*ex
p_polar(I*pi)/a)/(2*gamma(m/2 + 3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + a)^p*(g*x)^m, x)