\(\int (a+\frac {c e x}{f}+c x^2)^p (\frac {a f}{c}+e x+f x^2)^q \, dx\) [43]

Optimal result

Integrand size = 34, antiderivative size = 132 \[ \int \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^q \, dx=\frac {2^{-1-2 p-2 q} (e+2 f x) \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^q \left (-\frac {c f \left (\frac {a f}{c}+e x+f x^2\right )}{c e^2-4 a f^2}\right )^{-p-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-q,\frac {3}{2},\frac {c (e+2 f x)^2}{c e^2-4 a f^2}\right )}{f} \] Output:

2^(-1-2*p-2*q)*(2*f*x+e)*(a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q*(-c*f*(a* 
f/c+e*x+f*x^2)/(-4*a*f^2+c*e^2))^(-p-q)*hypergeom([1/2, -p-q],[3/2],c*(2*f 
*x+e)^2/(-4*a*f^2+c*e^2))/f
 

Mathematica [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.30 \[ \int \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^q \, dx=\frac {2^{-1+p+q} \left (\frac {a f}{c}+x (e+f x)\right )^q \left (a+\frac {c x (e+f x)}{f}\right )^p \left (-\sqrt {c e^2-4 a f^2}+\sqrt {c} (e+2 f x)\right ) \left (1+\frac {\sqrt {c} (e+2 f x)}{\sqrt {c e^2-4 a f^2}}\right )^{-p-q} \operatorname {Hypergeometric2F1}\left (-p-q,1+p+q,2+p+q,\frac {1}{2}-\frac {\sqrt {c} (e+2 f x)}{2 \sqrt {c e^2-4 a f^2}}\right )}{\sqrt {c} f (1+p+q)} \] Input:

Integrate[(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^q,x]
 

Output:

(2^(-1 + p + q)*((a*f)/c + x*(e + f*x))^q*(a + (c*x*(e + f*x))/f)^p*(-Sqrt 
[c*e^2 - 4*a*f^2] + Sqrt[c]*(e + 2*f*x))*(1 + (Sqrt[c]*(e + 2*f*x))/Sqrt[c 
*e^2 - 4*a*f^2])^(-p - q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p + q, 
1/2 - (Sqrt[c]*(e + 2*f*x))/(2*Sqrt[c*e^2 - 4*a*f^2])])/(Sqrt[c]*f*(1 + p 
+ q))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1296, 1096}

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.

\(\displaystyle \int \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^q \, dx\)

\(\Big \downarrow \) 1296

\(\displaystyle \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^{-p} \int \left (f x^2+e x+\frac {a f}{c}\right )^{p+q}dx\)

\(\Big \downarrow \) 1096

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

Input:

Int[(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^q,x]
 

Output:

-((2^(1 + p + q)*Sqrt[c]*(-((Sqrt[c]*(e - Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 
2*f*x))/Sqrt[c*e^2 - 4*a*f^2]))^(-1 - p - q)*(a + (c*e*x)/f + c*x^2)^p*((a 
*f)/c + e*x + f*x^2)^(1 + q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p + 
q, (Sqrt[c]*(e + Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 2*f*x))/(2*Sqrt[c*e^2 - 4 
*a*f^2])])/(Sqrt[c*e^2 - 4*a*f^2]*(1 + p + q)))
 

Defintions of rubi rules used

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1296

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_) 
^2)^(q_.), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x + c*x^2)^FracPart[p]/(d 
^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]))   Int[(d + e*x + f*x^2)^(p + q) 
, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[ 
b*d - a*e, 0] &&  !IntegerQ[p] &&  !IntegerQ[q] &&  !GtQ[c/f, 0]
 

Maple [F]

\[\int \left (a +\frac {c e x}{f}+c \,x^{2}\right )^{p} \left (\frac {a f}{c}+e x +f \,x^{2}\right )^{q}d x\]

Input:

int((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x)
 

Output:

int((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x)
 

Fricas [F]

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

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="fricas")
 

Output:

integral(((c*f*x^2 + c*e*x + a*f)/c)^q*((c*f*x^2 + c*e*x + a*f)/f)^p, x)
 

Sympy [F]

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

integrate((a+c*e*x/f+c*x**2)**p*(a*f/c+e*x+f*x**2)**q,x)
 

Output:

Integral((a + c*e*x/f + c*x**2)**p*(a*f/c + e*x + f*x**2)**q, x)
 

Maxima [F]

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

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="maxima")
 

Output:

integrate((c*x^2 + c*e*x/f + a)^p*(f*x^2 + e*x + a*f/c)^q, x)
 

Giac [F]

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

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="giac")
 

Output:

integrate((c*x^2 + c*e*x/f + a)^p*(f*x^2 + e*x + a*f/c)^q, x)
 

Mupad [F(-1)]

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

int((e*x + f*x^2 + (a*f)/c)^q*(a + c*x^2 + (c*e*x)/f)^p,x)
 

Output:

int((e*x + f*x^2 + (a*f)/c)^q*(a + c*x^2 + (c*e*x)/f)^p, x)
 

Reduce [F]

\[ \int \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^q \, dx =\text {Too large to display} \] Input:

int((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x)
 

Output:

(2*(a*f + c*e*x + c*f*x**2)**(p + q)*a*f + (a*f + c*e*x + c*f*x**2)**(p + 
q)*c*e*x - 8*int(((a*f + c*e*x + c*f*x**2)**(p + q)*x)/(2*a*f*p + 2*a*f*q 
+ a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2*c*f*q*x**2 + c*f* 
x**2),x)*a*c*f**2*p**2 - 16*int(((a*f + c*e*x + c*f*x**2)**(p + q)*x)/(2*a 
*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2*c* 
f*q*x**2 + c*f*x**2),x)*a*c*f**2*p*q - 4*int(((a*f + c*e*x + c*f*x**2)**(p 
 + q)*x)/(2*a*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f* 
p*x**2 + 2*c*f*q*x**2 + c*f*x**2),x)*a*c*f**2*p - 8*int(((a*f + c*e*x + c* 
f*x**2)**(p + q)*x)/(2*a*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e 
*x + 2*c*f*p*x**2 + 2*c*f*q*x**2 + c*f*x**2),x)*a*c*f**2*q**2 - 4*int(((a* 
f + c*e*x + c*f*x**2)**(p + q)*x)/(2*a*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2 
*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2*c*f*q*x**2 + c*f*x**2),x)*a*c*f**2*q + 
 2*int(((a*f + c*e*x + c*f*x**2)**(p + q)*x)/(2*a*f*p + 2*a*f*q + a*f + 2* 
c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2*c*f*q*x**2 + c*f*x**2),x)*c 
**2*e**2*p**2 + 4*int(((a*f + c*e*x + c*f*x**2)**(p + q)*x)/(2*a*f*p + 2*a 
*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2*c*f*q*x**2 + 
 c*f*x**2),x)*c**2*e**2*p*q + int(((a*f + c*e*x + c*f*x**2)**(p + q)*x)/(2 
*a*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*c*f*p*x**2 + 2* 
c*f*q*x**2 + c*f*x**2),x)*c**2*e**2*p + 2*int(((a*f + c*e*x + c*f*x**2)**( 
p + q)*x)/(2*a*f*p + 2*a*f*q + a*f + 2*c*e*p*x + 2*c*e*q*x + c*e*x + 2*...