\(\int (e x)^{-3-2 p} (c+d x) (a+b x^2)^p \, dx\) [328]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int (e x)^{-3-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {c (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a e (1+p)}-\frac {d (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{e^2 (1+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int (e x)^{-3-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (-\frac {a c+b c x^2}{a+a p}-\frac {2 d x \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )}{1+2 p}\right )}{2 e^3 x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, 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 = {545, 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.

\(\displaystyle \int (c+d x) (e x)^{-2 p-3} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 545

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

-1/2*(c*(a + b*x^2)^(1 + p))/(a*e*(1 + p)*(e*x)^(2*(1 + p))) - (d*(e*x)^(- 
1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(-1 - 2*p)/2, -p, (1 - 2*p)/2, -( 
(b*x^2)/a)])/(e^2*(1 + 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 545

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(-c)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), 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] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [F]

\[\int \left (e x \right )^{-3-2 p} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-3-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-3-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate((e*x)^(-3-2*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 - 3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 33.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int (e x)^{-3-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{- 2 p - 3} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} + \frac {a^{p} d e^{- 2 p - 3} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} \] Input:

integrate((e*x)**(-3-2*p)*(d*x+c)*(b*x**2+a)**p,x)
 

Output:

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

Maxima [F]

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

integrate((e*x)^(-3-2*p)*(d*x+c)*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(-3-2*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 - 3), x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 3),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x))/(e*x)^(2*p + 3), x)
 

Reduce [F]

\[ \int (e x)^{-3-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {-2 \left (b \,x^{2}+a \right )^{p} a c p -\left (b \,x^{2}+a \right )^{p} a c -2 \left (b \,x^{2}+a \right )^{p} a d p x -2 \left (b \,x^{2}+a \right )^{p} a d x -2 \left (b \,x^{2}+a \right )^{p} b c p \,x^{2}-\left (b \,x^{2}+a \right )^{p} b c \,x^{2}+8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p +x^{2 p} a +2 x^{2 p} b p \,x^{2}+x^{2 p} b \,x^{2}}d x \right ) a b d \,p^{3} x^{2}+12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p +x^{2 p} a +2 x^{2 p} b p \,x^{2}+x^{2 p} b \,x^{2}}d x \right ) a b d \,p^{2} x^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{2 x^{2 p} a p +x^{2 p} a +2 x^{2 p} b p \,x^{2}+x^{2 p} b \,x^{2}}d x \right ) a b d p \,x^{2}}{2 x^{2 p} e^{2 p} a \,e^{3} x^{2} \left (2 p^{2}+3 p +1\right )} \] Input:

int((e*x)^(-3-2*p)*(d*x+c)*(b*x^2+a)^p,x)
 

Output:

( - 2*(a + b*x**2)**p*a*c*p - (a + b*x**2)**p*a*c - 2*(a + b*x**2)**p*a*d* 
p*x - 2*(a + b*x**2)**p*a*d*x - 2*(a + b*x**2)**p*b*c*p*x**2 - (a + b*x**2 
)**p*b*c*x**2 + 8*x**(2*p)*int((a + b*x**2)**p/(2*x**(2*p)*a*p + x**(2*p)* 
a + 2*x**(2*p)*b*p*x**2 + x**(2*p)*b*x**2),x)*a*b*d*p**3*x**2 + 12*x**(2*p 
)*int((a + b*x**2)**p/(2*x**(2*p)*a*p + x**(2*p)*a + 2*x**(2*p)*b*p*x**2 + 
 x**(2*p)*b*x**2),x)*a*b*d*p**2*x**2 + 4*x**(2*p)*int((a + b*x**2)**p/(2*x 
**(2*p)*a*p + x**(2*p)*a + 2*x**(2*p)*b*p*x**2 + x**(2*p)*b*x**2),x)*a*b*d 
*p*x**2)/(2*x**(2*p)*e**(2*p)*a*e**3*x**2*(2*p**2 + 3*p + 1))