\(\int (e x)^{-1-2 p} (c+d x) (a+b x^2)^p \, dx\) [1890]

Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int (e x)^{-1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {d (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{e^2 (1-2 p)}-\frac {c (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e p} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int (e x)^{-1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 d p x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+c (-1+2 p) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )}{2 e p (-1+2 p)} \] Input:

Integrate[(e*x)^(-1 - 2*p)*(c + d*x)*(a + b*x^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {557, 279, 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.

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

\(\Big \downarrow \) 557

\(\displaystyle c \int (e x)^{-2 p-1} \left (b x^2+a\right )^pdx+\frac {d \int (e x)^{-2 p} \left (b x^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 279

\(\displaystyle c \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-2 p-1} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{e}\)

\(\Big \downarrow \) 278

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

Input:

Int[(e*x)^(-1 - 2*p)*(c + d*x)*(a + b*x^2)^p,x]
 

Output:

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

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], 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]
 

Maple [F]

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

Input:

int((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((d*x + c)*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 30.73 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int (e x)^{-1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{- 2 p - 1} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} + \frac {a^{p} d e^{- 2 p - 1} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right )} \] Input:

integrate((e*x)**(-1-2*p)*(d*x+c)*(b*x**2+a)**p,x)
 

Output:

a**p*c*e**(-2*p - 1)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_polar( 
I*pi)/a)/(2*x**(2*p)*gamma(1 - p)) + a**p*d*e**(-2*p - 1)*x**(1 - 2*p)*gam 
ma(1/2 - p)*hyper((-p, 1/2 - p), (3/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2* 
gamma(3/2 - p))
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Giac [F]

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

integrate((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((d*x + c)*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 1),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 1), x)
 

Reduce [F]

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

int((e*x)^(-1-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

((a + b*x**2)**p*d*x + 2*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2* 
p)*b*x**2),x)*a*d*p + x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*x),x)*c)/(x** 
(2*p)*e**(2*p)*e)