\(\int (e x)^{-2 p} (c+d x) (a+b x^2)^p \, dx\) [347]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int (e x)^{-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {c (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{e (1-2 p)}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{2 e^2 (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int (e x)^{-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {x (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 c (-1+p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+d (-1+2 p) x \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )\right )}{2 (-1+p) (-1+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 27.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.60 \[ \int (e x)^{-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} d e^{- 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} + \sqrt {b} b^{p - \frac {1}{2}} c e^{- 2 p} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )} \] Input:

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

Output:

a**p*d*x**(2 - 2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), b*x**2*exp_p 
olar(I*pi)/a)/(2*e**(2*p)*gamma(2 - p)) + sqrt(b)*b**(p - 1/2)*c*x*hyper(( 
-1/2, -p), (1/2,), a*exp_polar(I*pi)/(b*x**2))/e**(2*p)
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {2 \left (b \,x^{2}+a \right )^{p} c x +\left (b \,x^{2}+a \right )^{p} d \,x^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a c p +2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a d p}{2 x^{2 p} e^{2 p}} \] Input:

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

Output:

(2*(a + b*x**2)**p*c*x + (a + b*x**2)**p*d*x**2 + 4*x**(2*p)*int((a + b*x* 
*2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*c*p + 2*x**(2*p)*int(((a + b*x* 
*2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*d*p)/(2*x**(2*p)*e**(2*p))