\(\int \frac {(e x)^m (1-a^2 x^2)^p}{1+a x} \, dx\) [1296]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-p,\frac {3+m}{2},a^2 x^2\right )}{e (1+m)}-\frac {a (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},1-p,\frac {4+m}{2},a^2 x^2\right )}{e^2 (2+m)} \] Output:

(e*x)^(1+m)*hypergeom([1-p, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/e/(1+m)-a*(e*x 
)^(2+m)*hypergeom([1-p, 1+1/2*m],[2+1/2*m],a^2*x^2)/e^2/(2+m)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=x (e x)^m \left (-\frac {a x \operatorname {Hypergeometric2F1}\left (1+\frac {m}{2},1-p,2+\frac {m}{2},a^2 x^2\right )}{2+m}+\frac {\operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-p,\frac {3+m}{2},a^2 x^2\right )}{1+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {583, 557, 278}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 583

\(\displaystyle \int (1-a x) \left (1-a^2 x^2\right )^{p-1} (e x)^mdx\)

\(\Big \downarrow \) 557

\(\displaystyle \int (e x)^m \left (1-a^2 x^2\right )^{p-1}dx-\frac {a \int (e x)^{m+1} \left (1-a^2 x^2\right )^{p-1}dx}{e}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},1-p,\frac {m+3}{2},a^2 x^2\right )}{e (m+1)}-\frac {a (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},1-p,\frac {m+4}{2},a^2 x^2\right )}{e^2 (m+2)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 583

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, 0]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (-a^{2} x^{2}+1\right )^{p}}{a x +1}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{m}}{a x + 1} \,d x } \] Input:

integrate((e*x)^m*(-a^2*x^2+1)^p/(a*x+1),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.53 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.69 \[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\frac {0^{p} a^{m} a^{- m - 1} e^{m} m x^{m} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {m e^{i \pi }}{2}\right ) \Gamma \left (- \frac {m}{2}\right )}{4 \Gamma \left (1 - \frac {m}{2}\right )} - \frac {0^{p} a^{- m - 1} a^{m - 1} e^{m} m x^{m - 1} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} a^{- m - 1} a^{m - 1} e^{m} x^{m - 1} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {a^{2 p - 2} e^{m} p x^{m + 2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p + \frac {1}{2} \\ - \frac {m}{2} - p + \frac {3}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {3}{2}\right )} - \frac {a^{2 p - 1} e^{m} p x^{m + 2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac {m}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p \\ - \frac {m}{2} - p + 1 \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + 1\right )} \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{m}}{a x + 1} \,d x } \] Input:

integrate((e*x)^m*(-a^2*x^2+1)^p/(a*x+1),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{m}}{a x + 1} \,d x } \] Input:

integrate((e*x)^m*(-a^2*x^2+1)^p/(a*x+1),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (1-a^2\,x^2\right )}^p}{a\,x+1} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx=e^{m} \left (\int \frac {x^{m} \left (-a^{2} x^{2}+1\right )^{p}}{a x +1}d x \right ) \] Input:

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

Output:

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