\(\int (e x)^{-2-2 p} (c+d x) (a+b x^2)^p \, dx\) [1688]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int (e x)^{-2-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {c (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{e (1+2 p)}-\frac {d (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e^2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int (e x)^{-2-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=-\frac {x (e x)^{-2 (1+p)} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 c p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+d (1+2 p) x \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )}{2 p (1+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 139, 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.

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 24.33 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int (e x)^{-2-2 p} (c+d x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} c e^{- 2 p - 2} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {a^{p} d e^{- 2 p - 2} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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