\(\int (d+e x)^{-4-2 p} (a+b x+c x^2)^p \, dx\) [790]

Optimal result

Integrand size = 24, antiderivative size = 442 \[ \int (d+e x)^{-4-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (3+2 p)}-\frac {e (2 c d-b e) (2+p) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (1+p) (3+2 p)}+\frac {\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (a e+b d (3+2 p))\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^2 (1+2 p) (3+2 p)} \] Output:

-e*(e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)/(3+2*p)-1/2*e* 
(-b*e+2*c*d)*(2+p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^2/(p+1)/(3+2*p) 
/((e*x+d)^(2*p+2))+1/2*(b^2*e^2*(2+p)+2*c^2*d^2*(3+2*p)-2*c*e*(a*e+b*d*(3+ 
2*p)))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)*(e*x+d)^(-1-2*p)*(c*x^2+b*x+a)^p*hyper 
geom([-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)^2/(1+2*p)/(3+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 = 0.92 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.90 \[ \int (d+e x)^{-4-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {(d+e x)^{-3-2 p} (a+x (b+c x))^p \left (2 e (a+x (b+c x))+\frac {e (2 c d-b e) (2+p) (d+e x) (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right ) (1+p)}+\frac {\left (b^2 e^2 (2+p)+2 c^2 d^2 (3+2 p)-2 c e (a e+b d (3+2 p))\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}\right )^{-1-p} (d+e x)^2 \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}\right )}{\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2+e (-b d+a e)\right ) (1+2 p)}\right )}{2 \left (c d^2+e (-b d+a e)\right ) (3+2 p)} \] Input:

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

Output:

-1/2*((d + e*x)^(-3 - 2*p)*(a + x*(b + c*x))^p*(2*e*(a + x*(b + c*x)) + (e 
*(2*c*d - b*e)*(2 + p)*(d + e*x)*(a + x*(b + c*x)))/((c*d^2 + e*(-(b*d) + 
a*e))*(1 + p)) + ((b^2*e^2*(2 + p) + 2*c^2*d^2*(3 + 2*p) - 2*c*e*(a*e + b* 
d*(3 + 2*p)))*(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 + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)^2*Hypergeome 
tric2F1[-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)*(c*d^2 + e*(-(b*d) + a*e))*(1 + 2*p))))/((c*d^2 
 + e*(-(b*d) + a*e))*(3 + 2*p))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1167, 1228, 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.

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

\(\Big \downarrow \) 1167

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1155

\(\displaystyle -\frac {\frac {e (p+2) (2 c d-b e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )}-\frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}}{(2 p+3) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{(2 p+3) \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-((e*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e 
^2)*(3 + 2*p))) - ((e*(2*c*d - b*e)*(2 + p)*(a + b*x + c*x^2)^(1 + p))/(2* 
(c*d^2 - b*d*e + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) - ((b^2*e^2*(2 + p) 
 + 2*c^2*d^2*(3 + 2*p) - 2*c*e*(a*e + b*d*(3 + 2*p)))*(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 + 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))/((c*d^2 - b*d*e + a*e^2)*(3 
 + 2*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 1167

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(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, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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