\(\int (d x)^m (a+b x+c x^2)^p \, dx\) [784]

Optimal result

Integrand size = 18, antiderivative size = 137 \[ \int (d x)^m \left (a+b x+c x^2\right )^p \, dx=\frac {(d x)^{1+m} \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x+c x^2\right )^p \operatorname {AppellF1}\left (1+m,-p,-p,2+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{d (1+m)} \] Output:

(d*x)^(1+m)*(c*x^2+b*x+a)^p*AppellF1(1+m,-p,-p,2+m,-2*c*x/(b-(-4*a*c+b^2)^ 
(1/2)),-2*c*x/(b+(-4*a*c+b^2)^(1/2)))/d/(1+m)/((1+2*c*x/(b-(-4*a*c+b^2)^(1 
/2)))^p)/((1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.17 \[ \int (d x)^m \left (a+b x+c x^2\right )^p \, dx=\frac {x (d x)^m \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} (a+x (b+c x))^p \operatorname {AppellF1}\left (1+m,-p,-p,2+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )}{1+m} \] Input:

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

Output:

(x*(d*x)^m*(a + x*(b + c*x))^p*AppellF1[1 + m, -p, -p, 2 + m, (-2*c*x)/(b 
+ Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sq 
rt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c 
] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1179, 150}

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.

\(\displaystyle \int (d x)^m \left (a+b x+c x^2\right )^p \, dx\)

\(\Big \downarrow \) 1179

\(\displaystyle \frac {\left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p \int (d x)^m \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x}{b+\sqrt {b^2-4 a c}}+1\right )^pd(d x)}{d}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(d x)^{m+1} \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p \operatorname {AppellF1}\left (m+1,-p,-p,m+2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1)}\)

Input:

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

Output:

((d*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m, (-2*c*x) 
/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(d*(1 + m)*(1 
 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c]) 
)^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1179

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[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d x)^m \left (a+b x+c x^2\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

(d**m*(2*x**m*(a + b*x + c*x**2)**p*a*p + x**m*(a + b*x + c*x**2)**p*b*m*x 
 + x**m*(a + b*x + c*x**2)**p*b*p*x - 2*int((x**m*(a + b*x + c*x**2)**p*x) 
/(a*m**2 + 3*a*m*p + a*m + 2*a*p**2 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 
 2*b*p**2*x + b*p*x + c*m**2*x**2 + 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x** 
2 + c*p*x**2),x)*a*c*m**3*p - 10*int((x**m*(a + b*x + c*x**2)**p*x)/(a*m** 
2 + 3*a*m*p + a*m + 2*a*p**2 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 2*b*p* 
*2*x + b*p*x + c*m**2*x**2 + 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x**2 + c*p 
*x**2),x)*a*c*m**2*p**2 - 2*int((x**m*(a + b*x + c*x**2)**p*x)/(a*m**2 + 3 
*a*m*p + a*m + 2*a*p**2 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 2*b*p**2*x 
+ b*p*x + c*m**2*x**2 + 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x**2 + c*p*x**2 
),x)*a*c*m**2*p - 16*int((x**m*(a + b*x + c*x**2)**p*x)/(a*m**2 + 3*a*m*p 
+ a*m + 2*a*p**2 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 2*b*p**2*x + b*p*x 
 + c*m**2*x**2 + 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x**2 + c*p*x**2),x)*a* 
c*m*p**3 - 6*int((x**m*(a + b*x + c*x**2)**p*x)/(a*m**2 + 3*a*m*p + a*m + 
2*a*p**2 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 2*b*p**2*x + b*p*x + c*m** 
2*x**2 + 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x**2 + c*p*x**2),x)*a*c*m*p**2 
 - 8*int((x**m*(a + b*x + c*x**2)**p*x)/(a*m**2 + 3*a*m*p + a*m + 2*a*p**2 
 + a*p + b*m**2*x + 3*b*m*p*x + b*m*x + 2*b*p**2*x + b*p*x + c*m**2*x**2 + 
 3*c*m*p*x**2 + c*m*x**2 + 2*c*p**2*x**2 + c*p*x**2),x)*a*c*p**4 - 4*int(( 
x**m*(a + b*x + c*x**2)**p*x)/(a*m**2 + 3*a*m*p + a*m + 2*a*p**2 + a*p ...