\(\int (d x)^m (a c+(b c+a d) x+b d x^2)^p \, dx\) [173]

Optimal result

Integrand size = 27, antiderivative size = 88 \[ \int (d x)^m \left (a c+(b c+a d) x+b d x^2\right )^p \, dx=\frac {(d x)^{1+m} \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {d x}{c}\right )^{-p} \left (a c+(b c+a d) x+b d x^2\right )^p \operatorname {AppellF1}\left (1+m,-p,-p,2+m,-\frac {d x}{c},-\frac {b x}{a}\right )}{d (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int (d x)^m \left (a c+(b c+a d) x+b d x^2\right )^p \, dx=\frac {x (d x)^m \left (\frac {a+b x}{a}\right )^{-p} \left (\frac {c+d x}{c}\right )^{-p} ((a+b x) (c+d x))^p \operatorname {AppellF1}\left (1+m,-p,-p,2+m,-\frac {b x}{a},-\frac {d x}{c}\right )}{1+m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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