\(\int (d+e x)^{-3-2 p} (a+b x+c x^2)^p \, dx\) [789]

Optimal result

Integrand size = 24, antiderivative size = 332 \[ \int (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (1+p)}+\frac {(2 c d-b e) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+2 p)} \] Output:

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

Time = 0.38 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.89 \[ \int (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^p \, dx=\frac {(d+e x)^{-2 (1+p)} (a+x (b+c x))^p \left (-\frac {e (a+x (b+c x))}{1+p}+\frac {(-2 c d+b e) \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}\right )^{-1-p} (d+e x) \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}\right )}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+2 p)}\right )}{2 \left (c d^2+e (-b d+a e)\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1157, 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.

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

\(\Big \downarrow \) 1157

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

\(\Big \downarrow \) 1155

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

Input:

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

Output:

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

rule 1157

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x+d)^(-3-2*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 - 3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((e*x+d)^(-3-2*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 - 3), x)
 

Giac [F]

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

integrate((e*x+d)^(-3-2*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 - 3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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