\(\int (e x)^{-5-2 p} (c+d x) (a+b x^2)^p \, dx\) [1685]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {b c (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a^2 e^3 (1+p) (2+p)}-\frac {c (e x)^{-2 (2+p)} \left (a+b x^2\right )^{1+p}}{2 a e (2+p)}-\frac {d (e x)^{-3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-p,\frac {1}{2} (-1-2 p),-\frac {b x^2}{a}\right )}{e^2 (3+2 p)} \] Output:

1/2*b*c*(b*x^2+a)^(p+1)/a^2/e^3/(p+1)/(2+p)/((e*x)^(2*p+2))-1/2*c*(b*x^2+a 
)^(p+1)/a/e/(2+p)/((e*x)^(4+2*p))-d*(e*x)^(-3-2*p)*(b*x^2+a)^p*hypergeom([ 
-p, -3/2-p],[-1/2-p],-b*x^2/a)/e^2/(3+2*p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.75 \[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c (3+2 p) \operatorname {Hypergeometric2F1}\left (-2-p,-p,-1-p,-\frac {b x^2}{a}\right )+2 d (2+p) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-p,-\frac {1}{2}-p,-\frac {b x^2}{a}\right )\right )}{2 e^5 (2+p) (3+2 p) x^4} \] Input:

Integrate[(e*x)^(-5 - 2*p)*(c + d*x)*(a + b*x^2)^p,x]
 

Output:

-1/2*((a + b*x^2)^p*(c*(3 + 2*p)*Hypergeometric2F1[-2 - p, -p, -1 - p, -(( 
b*x^2)/a)] + 2*d*(2 + p)*x*Hypergeometric2F1[-3/2 - p, -p, -1/2 - p, -((b* 
x^2)/a)]))/(e^5*(2 + p)*(3 + 2*p)*x^4*(e*x)^(2*p)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {557, 246, 242, 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.

\(\displaystyle \int (c+d x) (e x)^{-2 p-5} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 557

\(\displaystyle c \int (e x)^{-2 p-5} \left (b x^2+a\right )^pdx+\frac {d \int (e x)^{-2 (p+2)} \left (b x^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 246

\(\displaystyle c \left (-\frac {\int (e x)^{-2 p-5} \left (b x^2+a\right )^{p+1}dx}{a (p+1)}-\frac {(e x)^{-2 (p+2)} \left (a+b x^2\right )^{p+1}}{2 a e (p+1)}\right )+\frac {d \int (e x)^{-2 (p+2)} \left (b x^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

Int[(e*x)^(-5 - 2*p)*(c + d*x)*(a + b*x^2)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 242

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

rule 246

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

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]

\[\int \left (e x \right )^{-5-2 p} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-5-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-5-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-2*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 - 5), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 36.72 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.66 \[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=- \frac {a a^{p} c e^{- 2 p - 5} p x^{- 2 p - 4} \left (1 + \frac {b x^{2}}{a}\right )^{p + 2} \Gamma \left (- p - 2\right )}{2 a \Gamma \left (- p\right ) + 2 b x^{2} \Gamma \left (- p\right )} - \frac {a a^{p} c e^{- 2 p - 5} x^{- 2 p - 4} \left (1 + \frac {b x^{2}}{a}\right )^{p + 2} \Gamma \left (- p - 2\right )}{2 a \Gamma \left (- p\right ) + 2 b x^{2} \Gamma \left (- p\right )} + \frac {a^{p} b c e^{- 2 p - 5} x^{2} x^{- 2 p - 4} \left (1 + \frac {b x^{2}}{a}\right )^{p + 2} \Gamma \left (- p - 2\right )}{2 a \Gamma \left (- p\right ) + 2 b x^{2} \Gamma \left (- p\right )} + \frac {a^{p} d e^{- 2 p - 5} x^{- 2 p - 3} \Gamma \left (- p - \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {3}{2} \\ - p - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (- p - \frac {1}{2}\right )} \] Input:

integrate((e*x)**(-5-2*p)*(d*x+c)*(b*x**2+a)**p,x)
 

Output:

-a*a**p*c*e**(-2*p - 5)*p*x**(-2*p - 4)*(1 + b*x**2/a)**(p + 2)*gamma(-p - 
 2)/(2*a*gamma(-p) + 2*b*x**2*gamma(-p)) - a*a**p*c*e**(-2*p - 5)*x**(-2*p 
 - 4)*(1 + b*x**2/a)**(p + 2)*gamma(-p - 2)/(2*a*gamma(-p) + 2*b*x**2*gamm 
a(-p)) + a**p*b*c*e**(-2*p - 5)*x**2*x**(-2*p - 4)*(1 + b*x**2/a)**(p + 2) 
*gamma(-p - 2)/(2*a*gamma(-p) + 2*b*x**2*gamma(-p)) + a**p*d*e**(-2*p - 5) 
*x**(-2*p - 3)*gamma(-p - 3/2)*hyper((-p, -p - 3/2), (-p - 1/2,), b*x**2*e 
xp_polar(I*pi)/a)/(2*gamma(-p - 1/2))
 

Maxima [F]

\[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-2*p)*(d*x+c)*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

d*e^(-2*p - 5)*integrate(e^(p*log(b*x^2 + a) - 2*p*log(x))/x^4, x) + 1/2*( 
b^2*x^4 - a*b*p*x^2 - a^2*(p + 1))*c*e^(-2*p - 5)*e^(p*log(b*x^2 + a) - 2* 
p*log(x))/((p^2 + 3*p + 2)*a^2*x^4)
 

Giac [F]

\[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-2*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 - 5), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (c+d\,x\right )}{{\left (e\,x\right )}^{2\,p+5}} \,d x \] Input:

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 5),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 5), x)
 

Reduce [F]

\[ \int (e x)^{-5-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {-2 \left (b \,x^{2}+a \right )^{p} a^{2} c \,p^{2}-5 \left (b \,x^{2}+a \right )^{p} a^{2} c p -3 \left (b \,x^{2}+a \right )^{p} a^{2} c -2 \left (b \,x^{2}+a \right )^{p} a^{2} d \,p^{2} x -6 \left (b \,x^{2}+a \right )^{p} a^{2} d p x -4 \left (b \,x^{2}+a \right )^{p} a^{2} d x -2 \left (b \,x^{2}+a \right )^{p} a b c \,p^{2} x^{2}-3 \left (b \,x^{2}+a \right )^{p} a b c p \,x^{2}+2 \left (b \,x^{2}+a \right )^{p} b^{2} c p \,x^{4}+3 \left (b \,x^{2}+a \right )^{p} b^{2} c \,x^{4}+8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p \,x^{2}+3 x^{2 p} a \,x^{2}+2 x^{2 p} b p \,x^{4}+3 x^{2 p} b \,x^{4}}d x \right ) a^{2} b d \,p^{4} x^{4}+36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p \,x^{2}+3 x^{2 p} a \,x^{2}+2 x^{2 p} b p \,x^{4}+3 x^{2 p} b \,x^{4}}d x \right ) a^{2} b d \,p^{3} x^{4}+52 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p \,x^{2}+3 x^{2 p} a \,x^{2}+2 x^{2 p} b p \,x^{4}+3 x^{2 p} b \,x^{4}}d x \right ) a^{2} b d \,p^{2} x^{4}+24 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p \,x^{2}+3 x^{2 p} a \,x^{2}+2 x^{2 p} b p \,x^{4}+3 x^{2 p} b \,x^{4}}d x \right ) a^{2} b d p \,x^{4}}{2 x^{2 p} e^{2 p} a^{2} e^{5} x^{4} \left (2 p^{3}+9 p^{2}+13 p +6\right )} \] Input:

int((e*x)^(-5-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

( - 2*(a + b*x**2)**p*a**2*c*p**2 - 5*(a + b*x**2)**p*a**2*c*p - 3*(a + b* 
x**2)**p*a**2*c - 2*(a + b*x**2)**p*a**2*d*p**2*x - 6*(a + b*x**2)**p*a**2 
*d*p*x - 4*(a + b*x**2)**p*a**2*d*x - 2*(a + b*x**2)**p*a*b*c*p**2*x**2 - 
3*(a + b*x**2)**p*a*b*c*p*x**2 + 2*(a + b*x**2)**p*b**2*c*p*x**4 + 3*(a + 
b*x**2)**p*b**2*c*x**4 + 8*x**(2*p)*int((a + b*x**2)**p/(2*x**(2*p)*a*p*x* 
*2 + 3*x**(2*p)*a*x**2 + 2*x**(2*p)*b*p*x**4 + 3*x**(2*p)*b*x**4),x)*a**2* 
b*d*p**4*x**4 + 36*x**(2*p)*int((a + b*x**2)**p/(2*x**(2*p)*a*p*x**2 + 3*x 
**(2*p)*a*x**2 + 2*x**(2*p)*b*p*x**4 + 3*x**(2*p)*b*x**4),x)*a**2*b*d*p**3 
*x**4 + 52*x**(2*p)*int((a + b*x**2)**p/(2*x**(2*p)*a*p*x**2 + 3*x**(2*p)* 
a*x**2 + 2*x**(2*p)*b*p*x**4 + 3*x**(2*p)*b*x**4),x)*a**2*b*d*p**2*x**4 + 
24*x**(2*p)*int((a + b*x**2)**p/(2*x**(2*p)*a*p*x**2 + 3*x**(2*p)*a*x**2 + 
 2*x**(2*p)*b*p*x**4 + 3*x**(2*p)*b*x**4),x)*a**2*b*d*p*x**4)/(2*x**(2*p)* 
e**(2*p)*a**2*e**5*x**4*(2*p**3 + 9*p**2 + 13*p + 6))