\(\int (b x)^m (c+d x)^n (e+f x)^2 \, dx\) [571]

Optimal result

Integrand size = 20, antiderivative size = 165 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=-\frac {f (c f (2+m)-2 d e (3+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f^2 (b x)^{2+m} (c+d x)^{1+n}}{b^2 d (3+m+n)}+\frac {\left (\frac {e^2}{c+c m}+\frac {f (c f (2+m)-2 d e (3+m+n))}{d^2 (2+m+n) (3+m+n)}\right ) (b x)^{1+m} (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+m,-\frac {d x}{c}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.93 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {x (b x)^m (c+d x)^n \left (f (c+d x) (e+f x)-\frac {f (1+m) (c f (2+m)-d e (4+m+n)) (c+d x)+(d e (2+m+n) (c f (1+m)-d e (3+m+n))-c f (1+m) (c f (2+m)-d e (4+m+n))) \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{d (1+m) (2+m+n)}\right )}{d (3+m+n)} \] Input:

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

Output:

(x*(b*x)^m*(c + d*x)^n*(f*(c + d*x)*(e + f*x) - (f*(1 + m)*(c*f*(2 + m) - 
d*e*(4 + m + n))*(c + d*x) + ((d*e*(2 + m + n)*(c*f*(1 + m) - d*e*(3 + m + 
 n)) - c*f*(1 + m)*(c*f*(2 + m) - d*e*(4 + m + n)))*Hypergeometric2F1[1 + 
m, -n, 2 + m, -((d*x)/c)])/(1 + (d*x)/c)^n)/(d*(1 + m)*(2 + m + n))))/(d*( 
3 + m + n))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {101, 25, 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.

\(\displaystyle \int (b x)^m (e+f x)^2 (c+d x)^n \, dx\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {\int -b (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{b d (m+n+3)}+\frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\int b (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{b d (m+n+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\int (b x)^m (c+d x)^n (e (c f (m+1)-d e (m+n+3))+f (c f (m+2)-d e (m+n+4)) x)dx}{d (m+n+3)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \int (b x)^m (c+d x)^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\)

\(\Big \downarrow \) 76

\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {(c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \int (b x)^m \left (\frac {d x}{c}+1\right )^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)}-\frac {\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e (c f (m+1)-d e (m+n+3))-\frac {c f (m+1) (c f (m+2)-d e (m+n+4))}{d (m+n+2)}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{b (m+1)}+\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d (m+n+2)}}{d (m+n+3)}\)

Input:

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

Output:

(f*(b*x)^(1 + m)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(3 + m + n)) - ((f*(c*f 
*(2 + m) - d*e*(4 + m + n))*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + 
 n)) + ((e*(c*f*(1 + m) - d*e*(3 + m + n)) - (c*f*(1 + m)*(c*f*(2 + m) - d 
*e*(4 + m + n)))/(d*(2 + m + n)))*(b*x)^(1 + m)*(c + d*x)^n*Hypergeometric 
2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(b*(1 + m)*(1 + (d*x)/c)^n))/(d*(3 + m 
+ n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.73 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.78 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {b^{m} c^{n} e^{2} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 b^{m} c^{n} e f x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac {b^{m} c^{n} f^{2} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \] Input:

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

Output:

b**m*c**n*e**2*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*ex 
p_polar(I*pi)/c)/gamma(m + 2) + 2*b**m*c**n*e*f*x**(m + 2)*gamma(m + 2)*hy 
per((-n, m + 2), (m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3) + b**m*c**n 
*f**2*x**(m + 3)*gamma(m + 3)*hyper((-n, m + 3), (m + 4,), d*x*exp_polar(I 
*pi)/c)/gamma(m + 4)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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