\(\int (e x)^{-3-2 p} (c+d x)^2 (a+b x^2)^p \, dx\) [327]

Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int (e x)^{-3-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {c^2 (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a e (1+p)}-\frac {2 c d (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{e^2 (1+2 p)}-\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e^3 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.88 \[ \int (e x)^{-3-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (4 a c d p (1+p) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+(1+2 p) \left (c^2 p \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+a d^2 (1+p) x^2 \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )}{2 a e^3 p (1+p) (1+2 p) x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {560, 27, 279, 278, 358, 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.

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

\(\Big \downarrow \) 560

\(\displaystyle \int (e x)^{-2 p-3} \left (b x^2+a\right )^p \left (c^2+d^2 x^2\right )dx+\frac {\int 2 c d (e x)^{-2 (p+1)} \left (b x^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (e x)^{-2 p-3} \left (b x^2+a\right )^p \left (c^2+d^2 x^2\right )dx+\frac {2 c d \int (e x)^{-2 (p+1)} \left (b x^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 279

\(\displaystyle \int (e x)^{-2 p-3} \left (b x^2+a\right )^p \left (c^2+d^2 x^2\right )dx+\frac {2 c d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-2 (p+1)} \left (\frac {b x^2}{a}+1\right )^pdx}{e}\)

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 358

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 358

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

rule 560

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Module[{k}, Int[(e*x)^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2* 
k)*x^(2*k), {k, 0, n/2}]*(a + b*x^2)^p, x] + Simp[1/e   Int[(e*x)^(m + 1)*S 
um[Binomial[n, 2*k + 1]*c^(n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1) 
/2}]*(a + b*x^2)^p, x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && IGtQ[n, 1] & 
&  !IntegerQ[m] && EqQ[m + n + 2*p + 1, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 42.83 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.90 \[ \int (e x)^{-3-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{2} e^{- 2 p - 3} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} + \frac {a^{p} c d e^{- 2 p - 3} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {1}{2} - p\right )} + \frac {a^{p} d^{2} e^{- 2 p - 3} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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