Integrand size = 19, antiderivative size = 76 \[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\frac {x \left (1+\frac {b x}{a}\right )^{-p} (c+d x)^q \left (1+\frac {d x}{c}\right )^{-q} \left (a x+b x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,-q,2+p,-\frac {b x}{a},-\frac {d x}{c}\right )}{1+p} \] Output:
x*(d*x+c)^q*(b*x^2+a*x)^p*AppellF1(p+1,-p,-q,2+p,-b*x/a,-d*x/c)/(p+1)/((1+ b*x/a)^p)/((1+d*x/c)^q)
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\frac {x \left (\frac {a+b x}{a}\right )^{-p} (x (a+b x))^p (c+d x)^q \left (\frac {c+d x}{c}\right )^{-q} \operatorname {AppellF1}\left (1+p,-p,-q,2+p,-\frac {b x}{a},-\frac {d x}{c}\right )}{1+p} \] Input:
Integrate[(c + d*x)^q*(a*x + b*x^2)^p,x]
Output:
(x*(x*(a + b*x))^p*(c + d*x)^q*AppellF1[1 + p, -p, -q, 2 + p, -((b*x)/a), -((d*x)/c)])/((1 + p)*((a + b*x)/a)^p*((c + d*x)/c)^q)
Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
| \(\displaystyle \int \left (a x+b x^2\right )^p (c+d x)^q \, dx\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^p \left (-\frac {d x}{c}\right )^{-p} \left (1-\frac {b (c+d x)}{b c-a d}\right )^{-p} \int (c+d x)^q \left (1-\frac {c+d x}{c}\right )^p \left (1-\frac {b (c+d x)}{b c-a d}\right )^pd(c+d x)}{d}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^p \left (-\frac {d x}{c}\right )^{-p} (c+d x)^{q+1} \left (1-\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {AppellF1}\left (q+1,-p,-p,q+2,\frac {c+d x}{c},\frac {b (c+d x)}{b c-a d}\right )}{d (q+1)}\) |
Input:
Int[(c + d*x)^q*(a*x + b*x^2)^p,x]
Output:
((c + d*x)^(1 + q)*(a*x + b*x^2)^p*AppellF1[1 + q, -p, -p, 2 + q, (c + d*x )/c, (b*(c + d*x))/(b*c - a*d)])/(d*(1 + q)*(-((d*x)/c))^p*(1 - (b*(c + d* x))/(b*c - a*d))^p)
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])
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]
\[\int \left (d x +c \right )^{q} \left (b \,x^{2}+a x \right )^{p}d x\]
Input:
int((d*x+c)^q*(b*x^2+a*x)^p,x)
Output:
int((d*x+c)^q*(b*x^2+a*x)^p,x)
\[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a x\right )}^{p} {\left (d x + c\right )}^{q} \,d x } \] Input:
integrate((d*x+c)^q*(b*x^2+a*x)^p,x, algorithm="fricas")
Output:
integral((b*x^2 + a*x)^p*(d*x + c)^q, x)
\[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\int \left (x \left (a + b x\right )\right )^{p} \left (c + d x\right )^{q}\, dx \] Input:
integrate((d*x+c)**q*(b*x**2+a*x)**p,x)
Output:
Integral((x*(a + b*x))**p*(c + d*x)**q, x)
\[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a x\right )}^{p} {\left (d x + c\right )}^{q} \,d x } \] Input:
integrate((d*x+c)^q*(b*x^2+a*x)^p,x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^p*(d*x + c)^q, x)
\[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a x\right )}^{p} {\left (d x + c\right )}^{q} \,d x } \] Input:
integrate((d*x+c)^q*(b*x^2+a*x)^p,x, algorithm="giac")
Output:
integrate((b*x^2 + a*x)^p*(d*x + c)^q, x)
Timed out. \[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\int {\left (b\,x^2+a\,x\right )}^p\,{\left (c+d\,x\right )}^q \,d x \] Input:
int((a*x + b*x^2)^p*(c + d*x)^q,x)
Output:
int((a*x + b*x^2)^p*(c + d*x)^q, x)
\[ \int (c+d x)^q \left (a x+b x^2\right )^p \, dx=\text {too large to display} \] Input:
int((d*x+c)^q*(b*x^2+a*x)^p,x)
Output:
((c + d*x)**q*(a*x + b*x**2)**p*a*c*p + (c + d*x)**q*(a*x + b*x**2)**p*a*c *q + (c + d*x)**q*(a*x + b*x**2)**p*a*d*p*x + (c + d*x)**q*(a*x + b*x**2)* *p*a*d*q*x + 2*(c + d*x)**q*(a*x + b*x**2)**p*b*c*p*x + 2*int(((c + d*x)** q*(a*x + b*x**2)**p*x)/(2*a**2*c*d*p**2 + 3*a**2*c*d*p*q + a**2*c*d*p + a* *2*c*d*q**2 + a**2*c*d*q + 2*a**2*d**2*p**2*x + 3*a**2*d**2*p*q*x + a**2*d **2*p*x + a**2*d**2*q**2*x + a**2*d**2*q*x + 4*a*b*c**2*p**2 + 2*a*b*c**2* p*q + 2*a*b*c**2*p + 6*a*b*c*d*p**2*x + 5*a*b*c*d*p*q*x + 3*a*b*c*d*p*x + a*b*c*d*q**2*x + a*b*c*d*q*x + 2*a*b*d**2*p**2*x**2 + 3*a*b*d**2*p*q*x**2 + a*b*d**2*p*x**2 + a*b*d**2*q**2*x**2 + a*b*d**2*q*x**2 + 4*b**2*c**2*p** 2*x + 2*b**2*c**2*p*q*x + 2*b**2*c**2*p*x + 4*b**2*c*d*p**2*x**2 + 2*b**2* c*d*p*q*x**2 + 2*b**2*c*d*p*x**2),x)*a**3*d**3*p**4 + 5*int(((c + d*x)**q* (a*x + b*x**2)**p*x)/(2*a**2*c*d*p**2 + 3*a**2*c*d*p*q + a**2*c*d*p + a**2 *c*d*q**2 + a**2*c*d*q + 2*a**2*d**2*p**2*x + 3*a**2*d**2*p*q*x + a**2*d** 2*p*x + a**2*d**2*q**2*x + a**2*d**2*q*x + 4*a*b*c**2*p**2 + 2*a*b*c**2*p* q + 2*a*b*c**2*p + 6*a*b*c*d*p**2*x + 5*a*b*c*d*p*q*x + 3*a*b*c*d*p*x + a* b*c*d*q**2*x + a*b*c*d*q*x + 2*a*b*d**2*p**2*x**2 + 3*a*b*d**2*p*q*x**2 + a*b*d**2*p*x**2 + a*b*d**2*q**2*x**2 + a*b*d**2*q*x**2 + 4*b**2*c**2*p**2* x + 2*b**2*c**2*p*q*x + 2*b**2*c**2*p*x + 4*b**2*c*d*p**2*x**2 + 2*b**2*c* d*p*q*x**2 + 2*b**2*c*d*p*x**2),x)*a**3*d**3*p**3*q + int(((c + d*x)**q*(a *x + b*x**2)**p*x)/(2*a**2*c*d*p**2 + 3*a**2*c*d*p*q + a**2*c*d*p + a**...