\(\int (e x)^{-4-2 p} (c+d x)^2 (a+b x^2)^p \, dx\) [1913]

Optimal result

Integrand size = 26, antiderivative size = 162 \[ \int (e x)^{-4-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {d^2 (e x)^{-3-2 p} \left (a+b x^2\right )^{1+p}}{b e}-\frac {c d (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{a e^2 (1+p)}-\frac {\left (b c^2-a d^2 (3+2 p)\right ) (e x)^{-3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-p,\frac {1}{2} (-1-2 p),-\frac {b x^2}{a}\right )}{b e (3+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06 \[ \int (e x)^{-4-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (a c^2 \left (1+3 p+2 p^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-p,-\frac {1}{2}-p,-\frac {b x^2}{a}\right )+d (3+2 p) x \left (c (1+2 p) \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+a d (1+p) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )\right )\right )}{a e^4 (1+p) (1+2 p) (3+2 p) x^3} \] Input:

Integrate[(e*x)^(-4 - 2*p)*(c + d*x)^2*(a + b*x^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {559, 25, 557, 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.

\(\displaystyle \int (c+d x)^2 (e x)^{-2 p-4} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

Int[(e*x)^(-4 - 2*p)*(c + d*x)^2*(a + b*x^2)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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]
 

rule 559

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-4-2*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 - 4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 59.87 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10 \[ \int (e x)^{-4-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{2} e^{- 2 p - 4} x^{- 2 p - 3} \Gamma \left (- p - \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {3}{2} \\ - p - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (- p - \frac {1}{2}\right )} + \frac {a^{p} c d e^{- 2 p - 4} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{\Gamma \left (- p\right )} + \frac {a^{p} d^{2} e^{- 2 p - 4} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} \] Input:

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

Output:

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

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

integrate((e*x)^(-4-2*p)*(d*x+c)^2*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^2*(b*x^2 + a)^p*(e*x)^(-2*p - 4), x)
 

Giac [F]

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

integrate((e*x)^(-4-2*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 - 4), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-4-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {2 \left (b \,x^{2}+a \right )^{p} a^{2} d^{2} p^{2}+2 \left (b \,x^{2}+a \right )^{p} a^{2} d^{2} p -\left (b \,x^{2}+a \right )^{p} a b \,c^{2} p -\left (b \,x^{2}+a \right )^{p} a b \,c^{2}-3 \left (b \,x^{2}+a \right )^{p} a b c d x -3 \left (b \,x^{2}+a \right )^{p} a b \,d^{2} p \,x^{2}-3 \left (b \,x^{2}+a \right )^{p} a b \,d^{2} x^{2}-3 \left (b \,x^{2}+a \right )^{p} b^{2} c d \,x^{3}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{3} d^{2} p^{3} x^{3}+10 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{3} d^{2} p^{2} x^{3}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{3} d^{2} p \,x^{3}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} b \,c^{2} p^{2} x^{3}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} b \,c^{2} p \,x^{3}}{3 x^{2 p} e^{2 p} a b \,e^{4} x^{3} \left (p +1\right )} \] Input:

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

Output:

(2*(a + b*x**2)**p*a**2*d**2*p**2 + 2*(a + b*x**2)**p*a**2*d**2*p - (a + b 
*x**2)**p*a*b*c**2*p - (a + b*x**2)**p*a*b*c**2 - 3*(a + b*x**2)**p*a*b*c* 
d*x - 3*(a + b*x**2)**p*a*b*d**2*p*x**2 - 3*(a + b*x**2)**p*a*b*d**2*x**2 
- 3*(a + b*x**2)**p*b**2*c*d*x**3 + 4*x**(2*p)*int((a + b*x**2)**p/(x**(2* 
p)*a*x**4 + x**(2*p)*b*x**6),x)*a**3*d**2*p**3*x**3 + 10*x**(2*p)*int((a + 
 b*x**2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**3*d**2*p**2*x**3 + 6 
*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**3* 
d**2*p*x**3 - 2*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b 
*x**6),x)*a**2*b*c**2*p**2*x**3 - 2*x**(2*p)*int((a + b*x**2)**p/(x**(2*p) 
*a*x**4 + x**(2*p)*b*x**6),x)*a**2*b*c**2*p*x**3)/(3*x**(2*p)*e**(2*p)*a*b 
*e**4*x**3*(p + 1))