\(\int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx\) [103]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx=\frac {f^2 F^{a+b c} x^m \Gamma (3+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^3 d^3 \log ^3(F)}-\frac {2 e f F^{a+b c} x^m \Gamma (2+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c} x^m \Gamma (1+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b d \log (F)} \] Output:

f^2*F^(b*c+a)*x^m*GAMMA(3+m,-b*d*x*ln(F))/b^3/d^3/ln(F)^3/((-b*d*x*ln(F))^ 
m)-2*e*f*F^(b*c+a)*x^m*GAMMA(2+m,-b*d*x*ln(F))/b^2/d^2/ln(F)^2/((-b*d*x*ln 
(F))^m)+e^2*F^(b*c+a)*x^m*GAMMA(1+m,-b*d*x*ln(F))/b/d/ln(F)/((-b*d*x*ln(F) 
)^m)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.62 \[ \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx=\frac {F^{a+b c} x^m (-b d x \log (F))^{-m} \left (f^2 \Gamma (3+m,-b d x \log (F))+b d e \log (F) (-2 f \Gamma (2+m,-b d x \log (F))+b d e \Gamma (1+m,-b d x \log (F)) \log (F))\right )}{b^3 d^3 \log ^3(F)} \] Input:

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

Output:

(F^(a + b*c)*x^m*(f^2*Gamma[3 + m, -(b*d*x*Log[F])] + b*d*e*Log[F]*(-2*f*G 
amma[2 + m, -(b*d*x*Log[F])] + b*d*e*Gamma[1 + m, -(b*d*x*Log[F])]*Log[F]) 
))/(b^3*d^3*Log[F]^3*(-(b*d*x*Log[F]))^m)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2629, 2009}

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.

\(\displaystyle \int x^m (e+f x)^2 F^{a+b (c+d x)} \, dx\)

\(\Big \downarrow \) 2629

\(\displaystyle \int \left (e^2 x^m F^{a+b (c+d x)}+2 e f x^{m+1} F^{a+b (c+d x)}+f^2 x^{m+2} F^{a+b (c+d x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {f^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \Gamma (m+3,-b d x \log (F))}{b^3 d^3 \log ^3(F)}-\frac {2 e f x^m F^{a+b c} (-b d x \log (F))^{-m} \Gamma (m+2,-b d x \log (F))}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \Gamma (m+1,-b d x \log (F))}{b d \log (F)}\)

Input:

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

Output:

(f^2*F^(a + b*c)*x^m*Gamma[3 + m, -(b*d*x*Log[F])])/(b^3*d^3*Log[F]^3*(-(b 
*d*x*Log[F]))^m) - (2*e*f*F^(a + b*c)*x^m*Gamma[2 + m, -(b*d*x*Log[F])])/( 
b^2*d^2*Log[F]^2*(-(b*d*x*Log[F]))^m) + (e^2*F^(a + b*c)*x^m*Gamma[1 + m, 
-(b*d*x*Log[F])])/(b*d*Log[F]*(-(b*d*x*Log[F]))^m)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(139)=278\).

Time = 0.56 (sec) , antiderivative size = 433, normalized size of antiderivative = 3.12

method	result	size
meijerg	\(-\frac {\ln \left (F \right )^{-3-m} \left (-b d \right )^{-m} F^{b c +a} f^{2} \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (m^{2}+3 m +2\right ) \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-m b d x \ln \left (F \right )+m^{2}-2 b d x \ln \left (F \right )+3 m +2\right ) {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (m^{2}+3 m +2\right ) \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b^{3} d^{3}}+\frac {2 \ln \left (F \right )^{-2-m} \left (-b d \right )^{-m} F^{b c +a} f e \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (1+m \right ) m \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}+x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (b d x \ln \left (F \right )-1-m \right ) {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} \left (1+m \right ) m \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b^{2} d^{2}}-\frac {F^{b c +a} \left (-b d \right )^{-m} \ln \left (F \right )^{-1-m} e^{2} \left (x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \Gamma \left (m \right ) \left (-b d x \ln \left (F \right )\right )^{-m}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} {\mathrm e}^{b d x \ln \left (F \right )}-x^{m} \left (-b d \right )^{m} \ln \left (F \right )^{m} m \left (-b d x \ln \left (F \right )\right )^{-m} \Gamma \left (m , -b d x \ln \left (F \right )\right )\right )}{b d}\)	\(433\)

Input:

int(F^(a+b*(d*x+c))*x^m*(f*x+e)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/b^3/d^3*ln(F)^(-3-m)*(-b*d)^(-m)*F^(b*c+a)*f^2*(x^m*(-b*d)^m*ln(F)^m*m* 
(m^2+3*m+2)*GAMMA(m)*(-b*d*x*ln(F))^(-m)-x^m*(-b*d)^m*ln(F)^m*(b^2*d^2*x^2 
*ln(F)^2-m*b*d*x*ln(F)+m^2-2*b*d*x*ln(F)+3*m+2)*exp(b*d*x*ln(F))-x^m*(-b*d 
)^m*ln(F)^m*m*(m^2+3*m+2)*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x*ln(F)))+2/b^2 
/d^2*ln(F)^(-2-m)*(-b*d)^(-m)*F^(b*c+a)*f*e*(x^m*(-b*d)^m*ln(F)^m*(1+m)*m* 
GAMMA(m)*(-b*d*x*ln(F))^(-m)+x^m*(-b*d)^m*ln(F)^m*(b*d*x*ln(F)-1-m)*exp(b* 
d*x*ln(F))-x^m*(-b*d)^m*ln(F)^m*(1+m)*m*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x 
*ln(F)))-F^(b*c+a)*(-b*d)^(-m)*ln(F)^(-1-m)*e^2/b/d*(x^m*(-b*d)^m*ln(F)^m* 
m*GAMMA(m)*(-b*d*x*ln(F))^(-m)-x^m*(-b*d)^m*ln(F)^m*exp(b*d*x*ln(F))-x^m*( 
-b*d)^m*ln(F)^m*m*(-b*d*x*ln(F))^(-m)*GAMMA(m,-b*d*x*ln(F)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.16 \[ \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx=-\frac {{\left ({\left (b d f^{2} m + 2 \, b d f^{2}\right )} x \log \left (F\right ) - {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} d^{2} e f x\right )} \log \left (F\right )^{2}\right )} F^{b d x + b c + a} x^{m} - {\left (b^{2} d^{2} e^{2} \log \left (F\right )^{2} + f^{2} m^{2} + 3 \, f^{2} m + 2 \, f^{2} - 2 \, {\left (b d e f m + b d e f\right )} \log \left (F\right )\right )} e^{\left (-m \log \left (-b d \log \left (F\right )\right ) + {\left (b c + a\right )} \log \left (F\right )\right )} \Gamma \left (m + 1, -b d x \log \left (F\right )\right )}{b^{3} d^{3} \log \left (F\right )^{3}} \] Input:

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

Output:

-(((b*d*f^2*m + 2*b*d*f^2)*x*log(F) - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x)* 
log(F)^2)*F^(b*d*x + b*c + a)*x^m - (b^2*d^2*e^2*log(F)^2 + f^2*m^2 + 3*f^ 
2*m + 2*f^2 - 2*(b*d*e*f*m + b*d*e*f)*log(F))*e^(-m*log(-b*d*log(F)) + (b* 
c + a)*log(F))*gamma(m + 1, -b*d*x*log(F)))/(b^3*d^3*log(F)^3)
 

Sympy [F]

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

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

Output:

Integral(F**(a + b*(c + d*x))*x**m*(e + f*x)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx=-\left (-b d x \log \left (F\right )\right )^{-m - 3} F^{b c + a} f^{2} x^{m + 3} \Gamma \left (m + 3, -b d x \log \left (F\right )\right ) - 2 \, \left (-b d x \log \left (F\right )\right )^{-m - 2} F^{b c + a} e f x^{m + 2} \Gamma \left (m + 2, -b d x \log \left (F\right )\right ) - \left (-b d x \log \left (F\right )\right )^{-m - 1} F^{b c + a} e^{2} x^{m + 1} \Gamma \left (m + 1, -b d x \log \left (F\right )\right ) \] Input:

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

Output:

-(-b*d*x*log(F))^(-m - 3)*F^(b*c + a)*f^2*x^(m + 3)*gamma(m + 3, -b*d*x*lo 
g(F)) - 2*(-b*d*x*log(F))^(-m - 2)*F^(b*c + a)*e*f*x^(m + 2)*gamma(m + 2, 
-b*d*x*log(F)) - (-b*d*x*log(F))^(-m - 1)*F^(b*c + a)*e^2*x^(m + 1)*gamma( 
m + 1, -b*d*x*log(F))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx=\frac {f^{b c +a} \left (x^{m} f^{b d x} \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e^{2}+2 x^{m} f^{b d x} \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e f x +x^{m} f^{b d x} \mathrm {log}\left (f \right )^{2} b^{2} d^{2} f^{2} x^{2}-2 x^{m} f^{b d x} \mathrm {log}\left (f \right ) b d e f m -2 x^{m} f^{b d x} \mathrm {log}\left (f \right ) b d e f -x^{m} f^{b d x} \mathrm {log}\left (f \right ) b d \,f^{2} m x -2 x^{m} f^{b d x} \mathrm {log}\left (f \right ) b d \,f^{2} x +x^{m} f^{b d x} f^{2} m^{2}+3 x^{m} f^{b d x} f^{2} m +2 x^{m} f^{b d x} f^{2}-\left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) \mathrm {log}\left (f \right )^{2} b^{2} d^{2} e^{2} m +2 \left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) \mathrm {log}\left (f \right ) b d e f \,m^{2}+2 \left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) \mathrm {log}\left (f \right ) b d e f m -\left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) f^{2} m^{3}-3 \left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) f^{2} m^{2}-2 \left (\int \frac {x^{m} f^{b d x}}{x}d x \right ) f^{2} m \right )}{\mathrm {log}\left (f \right )^{3} b^{3} d^{3}} \] Input:

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

Output:

(f**(a + b*c)*(x**m*f**(b*d*x)*log(f)**2*b**2*d**2*e**2 + 2*x**m*f**(b*d*x 
)*log(f)**2*b**2*d**2*e*f*x + x**m*f**(b*d*x)*log(f)**2*b**2*d**2*f**2*x** 
2 - 2*x**m*f**(b*d*x)*log(f)*b*d*e*f*m - 2*x**m*f**(b*d*x)*log(f)*b*d*e*f 
- x**m*f**(b*d*x)*log(f)*b*d*f**2*m*x - 2*x**m*f**(b*d*x)*log(f)*b*d*f**2* 
x + x**m*f**(b*d*x)*f**2*m**2 + 3*x**m*f**(b*d*x)*f**2*m + 2*x**m*f**(b*d* 
x)*f**2 - int((x**m*f**(b*d*x))/x,x)*log(f)**2*b**2*d**2*e**2*m + 2*int((x 
**m*f**(b*d*x))/x,x)*log(f)*b*d*e*f*m**2 + 2*int((x**m*f**(b*d*x))/x,x)*lo 
g(f)*b*d*e*f*m - int((x**m*f**(b*d*x))/x,x)*f**2*m**3 - 3*int((x**m*f**(b* 
d*x))/x,x)*f**2*m**2 - 2*int((x**m*f**(b*d*x))/x,x)*f**2*m))/(log(f)**3*b* 
*3*d**3)