\(\int (d+e x)^{-2-2 p} (a+b x+c x^2)^p \, dx\) [788]

Optimal result

Integrand size = 24, antiderivative size = 248 \[ \int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx=\frac {\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+2 p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.54 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00 \[ \int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {\left (\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )^{1-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} (d+e x)^{-1-2 p} (a+x (b+c x))^p \left (1-\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )^{2 p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}\right )}{e+2 e p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1155}

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.

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

\(\Big \downarrow \) 1155

\(\displaystyle \frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{(2 p+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 1155

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(b - q + 2*c*x))*(d + e*x)^ 
(m + 1)*((a + b*x + c*x^2)^p/((m + 1)*(2*c*d - b*e + e*q)*((2*c*d - b*e + e 
*q)*((b + q + 2*c*x)/((2*c*d - b*e - e*q)*(b - q + 2*c*x))))^p))*Hypergeome 
tric2F1[m + 1, -p, m + 2, -4*c*q*((d + e*x)/((2*c*d - b*e - e*q)*(b - q + 2 
*c*x)))], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[m + 2*p + 2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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