3.313 \(\int \frac{(g x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx\)

Optimal. Leaf size=89 \[ \frac{(g x)^{m+1} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};a^2 x^2\right )}{g (m+1)}-\frac{a (g x)^{m+2} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};a^2 x^2\right )}{g^2 (m+2)} \]

[Out]

((g*x)^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 - p, (3 + m)/2, a^2*x^2])/(g*(1 +
m)) - (a*(g*x)^(2 + m)*Hypergeometric2F1[(2 + m)/2, 1 - p, (4 + m)/2, a^2*x^2])/
(g^2*(2 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.211705, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{(g x)^{m+1} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};a^2 x^2\right )}{g (m+1)}-\frac{a (g x)^{m+2} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};a^2 x^2\right )}{g^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

((g*x)^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 - p, (3 + m)/2, a^2*x^2])/(g*(1 +
m)) - (a*(g*x)^(2 + m)*Hypergeometric2F1[(2 + m)/2, 1 - p, (4 + m)/2, a^2*x^2])/
(g^2*(2 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.6605, size = 63, normalized size = 0.71 \[ - \frac{a \left (g x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2}} \right )}}{g^{2} \left (m + 2\right )} + \frac{\left (g x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2}} \right )}}{g \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(-a**2*x**2+1)**p/(a*x+1),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.276596, size = 145, normalized size = 1.63 \[ \frac{(m+2) x (1-a x)^p (a x+1)^{p-1} (g x)^m F_1(m+1;-p,1-p;m+2;a x,-a x)}{(m+1) \left (a x \left ((p-1) F_1(m+2;-p,2-p;m+3;a x,-a x)-p \, _2F_1\left (\frac{m}{2}+1,1-p;\frac{m}{2}+2;a^2 x^2\right )\right )+(m+2) F_1(m+1;-p,1-p;m+2;a x,-a x)\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

((2 + m)*x*(g*x)^m*(1 - a*x)^p*(1 + a*x)^(-1 + p)*AppellF1[1 + m, -p, 1 - p, 2 +
m, a*x, -(a*x)])/((1 + m)*((2 + m)*AppellF1[1 + m, -p, 1 - p, 2 + m, a*x, -(a*x
)] + a*x*((-1 + p)*AppellF1[2 + m, -p, 2 - p, 3 + m, a*x, -(a*x)] - p*Hypergeome
tricPFQ[{1 + m/2, 1 - p}, {2 + m/2}, a^2*x^2])))

_______________________________________________________________________________________

Maple [F]  time = 0.405, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{p}}{ax+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((g*x)^m*(-a^2*x^2+1)^p/(a*x+1),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-a^2*x^2 + 1)^p*(g*x)^m/(a*x + 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-a^{2} x^{2} + 1\right )}^{p} \left (g x\right )^{m}}{a x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-a^2*x^2 + 1)^p*(g*x)^m/(a*x + 1), x)

_______________________________________________________________________________________

Sympy [A]  time = 42.96, size = 308, normalized size = 3.46 \[ \frac{0^{p} g^{m} m x^{m} \Phi \left (\frac{1}{a^{2} x^{2}}, 1, \frac{m e^{i \pi }}{2}\right ) \Gamma \left (- \frac{m}{2}\right )}{4 a \Gamma \left (- \frac{m}{2} + 1\right )} - \frac{0^{p} g^{m} m x^{m} \Phi \left (\frac{1}{a^{2} x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )}{4 a^{2} x \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right )} + \frac{0^{p} g^{m} x^{m} \Phi \left (\frac{1}{a^{2} x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )}{4 a^{2} x \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right )} - \frac{a^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - \frac{m}{2} - p \\ - \frac{m}{2} - p + 1 \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + 1\right )} + \frac{a^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - \frac{m}{2} - p + \frac{1}{2} \\ - \frac{m}{2} - p + \frac{3}{2} \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)**m*(-a**2*x**2+1)**p/(a*x+1),x)

[Out]

0**p*g**m*m*x**m*lerchphi(1/(a**2*x**2), 1, m*exp_polar(I*pi)/2)*gamma(-m/2)/(4*
a*gamma(-m/2 + 1)) - 0**p*g**m*m*x**m*lerchphi(1/(a**2*x**2), 1, -m/2 + 1/2)*gam
ma(-m/2 + 1/2)/(4*a**2*x*gamma(-m/2 + 3/2)) + 0**p*g**m*x**m*lerchphi(1/(a**2*x*
*2), 1, -m/2 + 1/2)*gamma(-m/2 + 1/2)/(4*a**2*x*gamma(-m/2 + 3/2)) - a**(2*p)*g*
*m*p*x**m*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p)*hyper((-p + 1, -m/2 - p)
, (-m/2 - p + 1,), 1/(a**2*x**2))/(2*a*gamma(p + 1)*gamma(-m/2 - p + 1)) + a**(2
*p)*g**m*p*x**m*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p + 1/2)*hyper((-p +
1, -m/2 - p + 1/2), (-m/2 - p + 3/2,), 1/(a**2*x**2))/(2*a**2*x*gamma(p + 1)*gam
ma(-m/2 - p + 3/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-a^2*x^2 + 1)^p*(g*x)^m/(a*x + 1), x)