\(\int (e x)^{1-2 p} (c+d x) (a+b x^2)^p \, dx\) [296]

Optimal result

Integrand size = 24, antiderivative size = 146 \[ \int (e x)^{1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {d (e x)^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),-p,\frac {1}{2} (5-2 p),-\frac {b x^2}{a}\right )}{e^2 (3-2 p)}+\frac {c (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{2 e (1-p)} \] Output:

d*(e*x)^(3-2*p)*(b*x^2+a)^p*hypergeom([-p, 3/2-p],[5/2-p],-b*x^2/a)/e^2/(3 
-2*p)/((1+b*x^2/a)^p)+1/2*c*(e*x)^(2-2*p)*(b*x^2+a)^p*hypergeom([-p, 1-p], 
[2-p],-b*x^2/a)/e/(1-p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.79 \[ \int (e x)^{1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c (-3+2 p) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+2 d (-1+p) x \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,-p,\frac {5}{2}-p,-\frac {b x^2}{a}\right )\right )}{2 (-1+p) (-3+2 p)} \] Input:

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

Output:

-1/2*(e*x^2*(a + b*x^2)^p*(c*(-3 + 2*p)*Hypergeometric2F1[1 - p, -p, 2 - p 
, -((b*x^2)/a)] + 2*d*(-1 + p)*x*Hypergeometric2F1[3/2 - p, -p, 5/2 - p, - 
((b*x^2)/a)]))/((-1 + p)*(-3 + 2*p)*(e*x)^(2*p)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {c (e x)^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{2 e (1-p)}+\frac {d (e x)^{3-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),-p,\frac {1}{2} (5-2 p),-\frac {b x^2}{a}\right )}{e^2 (3-2 p)}\)

Input:

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

Output:

(d*(e*x)^(3 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(3 - 2*p)/2, -p, (5 - 2 
*p)/2, -((b*x^2)/a)])/(e^2*(3 - 2*p)*(1 + (b*x^2)/a)^p) + (c*(e*x)^(2 - 2* 
p)*(a + b*x^2)^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^2)/a)])/(2*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 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 = 35.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71 \[ \int (e x)^{1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{1 - 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} + \frac {a^{p} d e^{1 - 2 p} x^{3 - 2 p} \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right )} \] Input:

integrate((e*x)**(1-2*p)*(d*x+c)*(b*x**2+a)**p,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**(3 - 
2*p)*gamma(3/2 - p)*hyper((-p, 3/2 - p), (5/2 - p,), b*x**2*exp_polar(I*pi 
)/a)/(2*gamma(5/2 - 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. \[ \int (e x)^{1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\int {\left (e\,x\right )}^{1-2\,p}\,{\left (b\,x^2+a\right )}^p\,\left (c+d\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^{1-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {e \left (4 \left (b \,x^{2}+a \right )^{p} a d p x +3 \left (b \,x^{2}+a \right )^{p} b c \,x^{2}+2 \left (b \,x^{2}+a \right )^{p} b d \,x^{3}+8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d \,p^{2}-4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d p +6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b c p \right )}{6 x^{2 p} e^{2 p} b} \] Input:

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

Output:

(e*(4*(a + b*x**2)**p*a*d*p*x + 3*(a + b*x**2)**p*b*c*x**2 + 2*(a + b*x**2 
)**p*b*d*x**3 + 8*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x* 
*2),x)*a**2*d*p**2 - 4*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p) 
*b*x**2),x)*a**2*d*p + 6*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))/(6*x**(2*p)*e**(2*p)*b)