\(\int (c+d x)^{-3-2 p} (e+f x)^2 (a+b x^2)^p \, dx\) [178]

Optimal result

Integrand size = 28, antiderivative size = 467 \[ \int (c+d x)^{-3-2 p} (e+f x)^2 \left (a+b x^2\right )^p \, dx=-\frac {(d e-c f)^2 (c+d x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 d \left (b c^2+a d^2\right ) (1+p)}-\frac {f^2 (c+d x)^{-2 p} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{2 d^3 p}-\frac {(d e-c f) \left (2 a d^2 f+b c (d e+c f)\right ) \left (\sqrt {-a}-\sqrt {b} x\right ) \left (-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (\sqrt {-a}+\sqrt {b} x\right )}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )^{-p} (c+d x)^{-1-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {b} (c+d x)}{\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {-a}-\sqrt {b} x\right )}\right )}{d^2 \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right ) (1+2 p)} \] Output:

-1/2*(-c*f+d*e)^2*(b*x^2+a)^(p+1)/d/(a*d^2+b*c^2)/(p+1)/((d*x+c)^(2*p+2))- 
1/2*f^2*(b*x^2+a)^p*AppellF1(-2*p,-p,-p,1-2*p,(d*x+c)/(c-(-a)^(1/2)*d/b^(1 
/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^3/p/((d*x+c)^(2*p))/((1-(d*x+c)/( 
c-(-a)^(1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)-(-c*f 
+d*e)*(2*a*d^2*f+b*c*(c*f+d*e))*((-a)^(1/2)-b^(1/2)*x)*(d*x+c)^(-1-2*p)*(b 
*x^2+a)^p*hypergeom([-p, -1-2*p],[-2*p],2*(-a)^(1/2)*b^(1/2)*(d*x+c)/(b^(1 
/2)*c-(-a)^(1/2)*d)/((-a)^(1/2)-b^(1/2)*x))/d^2/(b^(1/2)*c+(-a)^(1/2)*d)/( 
a*d^2+b*c^2)/(1+2*p)/((-(b^(1/2)*c+(-a)^(1/2)*d)*((-a)^(1/2)+b^(1/2)*x)/(b 
^(1/2)*c-(-a)^(1/2)*d)/((-a)^(1/2)-b^(1/2)*x))^p)
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 744

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 744

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, 
 x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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