\(\int (e x)^{1-2 p} (a+b x^2)^p (c+d x^2) \, dx\) [1609]

Optimal result

Integrand size = 26, antiderivative size = 111 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{4 b e}-\frac {\left (\frac {a d}{b}-\frac {2 c}{1-p}\right ) (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{4 e} \] Output:

1/4*d*(e*x)^(2-2*p)*(b*x^2+a)^(p+1)/b/e-1/4*(a*d/b-2*c/(-p+1))*(e*x)^(2-2* 
p)*(b*x^2+a)^p*hypergeom([-p, -p+1],[2-p],-b*x^2/a)/e/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.97 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=-\frac {e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c (-2+p) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+d (-1+p) x^2 \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )\right )}{2 (-2+p) (-1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {363, 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.

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

\(\Big \downarrow \) 363

\(\displaystyle \frac {(2 b c-a d (1-p)) \int (e x)^{1-2 p} \left (b x^2+a\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{4 b e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} (2 b c-a d (1-p)) \int (e x)^{1-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{4 b e}\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

(d*(e*x)^(2 - 2*p)*(a + b*x^2)^(1 + p))/(4*b*e) + ((2*b*c - a*d*(1 - p))*( 
e*x)^(2 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^2) 
/a)])/(4*b*e*(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 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^2 + 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 = 46.51 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {a^{p} c e^{1 - 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} + \frac {a^{p} d e^{1 - 2 p} x^{4 - 2 p} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (3 - p\right )} \] Input:

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

Output:

a**p*c*e**(1 - 2*p)*x**(2 - 2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), 
 b*x**2*exp_polar(I*pi)/a)/(2*gamma(2 - p)) + a**p*d*e**(1 - 2*p)*x**(4 - 
2*p)*gamma(2 - p)*hyper((-p, 2 - p), (3 - p,), b*x**2*exp_polar(I*pi)/a)/( 
2*gamma(3 - p))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {e \left (\left (b \,x^{2}+a \right )^{p} a d p \,x^{2}+2 \left (b \,x^{2}+a \right )^{p} b c \,x^{2}+\left (b \,x^{2}+a \right )^{p} b d \,x^{4}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d \,p^{2}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d p +4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b c p \right )}{4 x^{2 p} e^{2 p} b} \] Input:

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

Output:

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