\(\int (g x)^n (d+e x)^m (a+b x+c x^2) \, dx\) [131]

Optimal result

Integrand size = 23, antiderivative size = 159 \[ \int (g x)^n (d+e x)^m \left (a+b x+c x^2\right ) \, dx=-\frac {(c d (2+n)-b e (3+m+n)) (g x)^{1+n} (d+e x)^{1+m}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+n} (d+e x)^{1+m}}{e g^2 (3+m+n)}+\frac {\left (\frac {a}{d+d n}+\frac {c d (2+n)-b e (3+m+n)}{e^2 (2+m+n) (3+m+n)}\right ) (g x)^{1+n} (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+n,-\frac {e x}{d}\right )}{g} \] Output:

-(c*d*(2+n)-b*e*(3+m+n))*(g*x)^(1+n)*(e*x+d)^(1+m)/e^2/g/(2+m+n)/(3+m+n)+c 
*(g*x)^(2+n)*(e*x+d)^(1+m)/e/g^2/(3+m+n)+(a/(d*n+d)+(c*d*(2+n)-b*e*(3+m+n) 
)/e^2/(2+m+n)/(3+m+n))*(g*x)^(1+n)*(e*x+d)^(1+m)*hypergeom([1, 2+m+n],[2+n 
],-e*x/d)/g
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int (g x)^n (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {x (g x)^n (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m} \left (c d^2 \operatorname {Hypergeometric2F1}\left (-2-m,1+n,2+n,-\frac {e x}{d}\right )+d (-2 c d+b e) \operatorname {Hypergeometric2F1}\left (-1-m,1+n,2+n,-\frac {e x}{d}\right )+\left (c d^2-b d e+a e^2\right ) \operatorname {Hypergeometric2F1}\left (-m,1+n,2+n,-\frac {e x}{d}\right )\right )}{e^2 (1+n)} \] Input:

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

Output:

(x*(g*x)^n*(d + e*x)^m*(c*d^2*Hypergeometric2F1[-2 - m, 1 + n, 2 + n, -((e 
*x)/d)] + d*(-2*c*d + b*e)*Hypergeometric2F1[-1 - m, 1 + n, 2 + n, -((e*x) 
/d)] + (c*d^2 - b*d*e + a*e^2)*Hypergeometric2F1[-m, 1 + n, 2 + n, -((e*x) 
/d)]))/(e^2*(1 + n)*(1 + (e*x)/d)^m)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1194, 27, 90, 76, 74}

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.

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

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\int g^2 (g x)^n (d+e x)^m (a e (m+n+3)-(c d (n+2)-b e (m+n+3)) x)dx}{e g^2 (m+n+3)}+\frac {c (g x)^{n+2} (d+e x)^{m+1}}{e g^2 (m+n+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (g x)^n (d+e x)^m (a e (m+n+3)-(c d (n+2)-b e (m+n+3)) x)dx}{e (m+n+3)}+\frac {c (g x)^{n+2} (d+e x)^{m+1}}{e g^2 (m+n+3)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\left (a e (m+n+3)+\frac {d (n+1) (c d (n+2)-b e (m+n+3))}{e (m+n+2)}\right ) \int (g x)^n (d+e x)^mdx-\frac {(g x)^{n+1} (d+e x)^{m+1} (c d (n+2)-b e (m+n+3))}{e g (m+n+2)}}{e (m+n+3)}+\frac {c (g x)^{n+2} (d+e x)^{m+1}}{e g^2 (m+n+3)}\)

\(\Big \downarrow \) 76

\(\displaystyle \frac {(d+e x)^m \left (\frac {e x}{d}+1\right )^{-m} \left (a e (m+n+3)+\frac {d (n+1) (c d (n+2)-b e (m+n+3))}{e (m+n+2)}\right ) \int (g x)^n \left (\frac {e x}{d}+1\right )^mdx-\frac {(g x)^{n+1} (d+e x)^{m+1} (c d (n+2)-b e (m+n+3))}{e g (m+n+2)}}{e (m+n+3)}+\frac {c (g x)^{n+2} (d+e x)^{m+1}}{e g^2 (m+n+3)}\)

\(\Big \downarrow \) 74

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

Input:

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

Output:

(c*(g*x)^(2 + n)*(d + e*x)^(1 + m))/(e*g^2*(3 + m + n)) + (-(((c*d*(2 + n) 
 - b*e*(3 + m + n))*(g*x)^(1 + n)*(d + e*x)^(1 + m))/(e*g*(2 + m + n))) + 
((a*e*(3 + m + n) + (d*(1 + n)*(c*d*(2 + n) - b*e*(3 + m + n)))/(e*(2 + m 
+ n)))*(g*x)^(1 + n)*(d + e*x)^m*Hypergeometric2F1[-m, 1 + n, 2 + n, -((e* 
x)/d)])/(g*(1 + n)*(1 + (e*x)/d)^m))/(e*(3 + m + n))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

Maple [F]

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

Input:

int((g*x)^n*(e*x+d)^m*(c*x^2+b*x+a),x)
 

Output:

int((g*x)^n*(e*x+d)^m*(c*x^2+b*x+a),x)
 

Fricas [F]

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

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

Output:

integral((c*x^2 + b*x + a)*(e*x + d)^m*(g*x)^n, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int (g x)^n (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {a d^{m} g^{n} x^{n + 1} \Gamma \left (n + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, n + 1 \\ n + 2 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (n + 2\right )} + \frac {b d^{m} g^{n} x^{n + 2} \Gamma \left (n + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, n + 2 \\ n + 3 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (n + 3\right )} + \frac {c d^{m} g^{n} x^{n + 3} \Gamma \left (n + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, n + 3 \\ n + 4 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (n + 4\right )} \] Input:

integrate((g*x)**n*(e*x+d)**m*(c*x**2+b*x+a),x)
 

Output:

a*d**m*g**n*x**(n + 1)*gamma(n + 1)*hyper((-m, n + 1), (n + 2,), e*x*exp_p 
olar(I*pi)/d)/gamma(n + 2) + b*d**m*g**n*x**(n + 2)*gamma(n + 2)*hyper((-m 
, n + 2), (n + 3,), e*x*exp_polar(I*pi)/d)/gamma(n + 3) + c*d**m*g**n*x**( 
n + 3)*gamma(n + 3)*hyper((-m, n + 3), (n + 4,), e*x*exp_polar(I*pi)/d)/ga 
mma(n + 4)
 

Maxima [F]

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

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

Output:

integrate((c*x^2 + b*x + a)*(e*x + d)^m*(g*x)^n, x)
 

Giac [F]

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

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

Output:

integrate((c*x^2 + b*x + a)*(e*x + d)^m*(g*x)^n, x)
 

Mupad [F(-1)]

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

int((g*x)^n*(d + e*x)^m*(a + b*x + c*x^2),x)
 

Output:

int((g*x)^n*(d + e*x)^m*(a + b*x + c*x^2), x)
 

Reduce [F]

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

int((g*x)^n*(e*x+d)^m*(c*x^2+b*x+a),x)
 

Output:

(g**n*(x**n*(d + e*x)**m*a*d*e**2*m**3 + 2*x**n*(d + e*x)**m*a*d*e**2*m**2 
*n + 5*x**n*(d + e*x)**m*a*d*e**2*m**2 + x**n*(d + e*x)**m*a*d*e**2*m*n**2 
 + 5*x**n*(d + e*x)**m*a*d*e**2*m*n + 6*x**n*(d + e*x)**m*a*d*e**2*m + x** 
n*(d + e*x)**m*a*e**3*m**3*x + 3*x**n*(d + e*x)**m*a*e**3*m**2*n*x + 5*x** 
n*(d + e*x)**m*a*e**3*m**2*x + 3*x**n*(d + e*x)**m*a*e**3*m*n**2*x + 10*x* 
*n*(d + e*x)**m*a*e**3*m*n*x + 6*x**n*(d + e*x)**m*a*e**3*m*x + x**n*(d + 
e*x)**m*a*e**3*n**3*x + 5*x**n*(d + e*x)**m*a*e**3*n**2*x + 6*x**n*(d + e* 
x)**m*a*e**3*n*x - x**n*(d + e*x)**m*b*d**2*e*m**2*n - x**n*(d + e*x)**m*b 
*d**2*e*m**2 - x**n*(d + e*x)**m*b*d**2*e*m*n**2 - 4*x**n*(d + e*x)**m*b*d 
**2*e*m*n - 3*x**n*(d + e*x)**m*b*d**2*e*m + x**n*(d + e*x)**m*b*d*e**2*m* 
*3*x + 2*x**n*(d + e*x)**m*b*d*e**2*m**2*n*x + 3*x**n*(d + e*x)**m*b*d*e** 
2*m**2*x + x**n*(d + e*x)**m*b*d*e**2*m*n**2*x + 3*x**n*(d + e*x)**m*b*d*e 
**2*m*n*x + x**n*(d + e*x)**m*b*e**3*m**3*x**2 + 3*x**n*(d + e*x)**m*b*e** 
3*m**2*n*x**2 + 4*x**n*(d + e*x)**m*b*e**3*m**2*x**2 + 3*x**n*(d + e*x)**m 
*b*e**3*m*n**2*x**2 + 8*x**n*(d + e*x)**m*b*e**3*m*n*x**2 + 3*x**n*(d + e* 
x)**m*b*e**3*m*x**2 + x**n*(d + e*x)**m*b*e**3*n**3*x**2 + 4*x**n*(d + e*x 
)**m*b*e**3*n**2*x**2 + 3*x**n*(d + e*x)**m*b*e**3*n*x**2 + x**n*(d + e*x) 
**m*c*d**3*m*n**2 + 3*x**n*(d + e*x)**m*c*d**3*m*n + 2*x**n*(d + e*x)**m*c 
*d**3*m - x**n*(d + e*x)**m*c*d**2*e*m**2*n*x - 2*x**n*(d + e*x)**m*c*d**2 
*e*m**2*x - x**n*(d + e*x)**m*c*d**2*e*m*n**2*x - 2*x**n*(d + e*x)**m*c...