\(\int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx\) [151]

Optimal result

Integrand size = 20, antiderivative size = 402 \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=\frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3^{-1-m} a^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a^3 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}-\frac {15 a^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a^3 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3^{-1-m} a^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \] Output:

5/2*a^3*(d*x+c)^(1+m)/d/(1+m)+1/8*3^(-1-m)*a^3*exp(3*e-3*c*f/d)*(d*x+c)^m* 
GAMMA(1+m,-3*f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)+3*2^(-3-m)*a^3*exp(2*e-2*c* 
f/d)*(d*x+c)^m*GAMMA(1+m,-2*f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)+15/8*a^3*exp 
(e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)-15/8*a^3* 
exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)-3*2^(-3 
-m)*a^3*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAMMA(1+m,2*f*(d*x+c)/d)/f/((f*(d*x+c) 
/d)^m)-1/8*3^(-1-m)*a^3*exp(-3*e+3*c*f/d)*(d*x+c)^m*GAMMA(1+m,3*f*(d*x+c)/ 
d)/f/((f*(d*x+c)/d)^m)
 

Mathematica [A] (warning: unable to verify)

Time = 1.52 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.07 \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=-\frac {2^{-6-m} 3^{-1-m} a^3 e^{-3 \left (e+\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} (1+\cosh (e+f x))^3 \left (-2^m d e^{6 e} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )-3^{2+m} d e^{5 e+\frac {c f}{d}} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )-5\ 2^m 3^{2+m} d e^{4 e+\frac {2 c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )+5\ 2^m 3^{2+m} d e^{2 e+\frac {4 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )+3^{2+m} d e^{e+\frac {5 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )+2^m e^{\frac {3 c f}{d}} \left (-20 3^{1+m} e^{3 e} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+d e^{\frac {3 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )\right )\right ) \text {sech}^6\left (\frac {1}{2} (e+f x)\right )}{d f (1+m)} \] Input:

Integrate[(c + d*x)^m*(a + a*Cosh[e + f*x])^3,x]
 

Output:

-((2^(-6 - m)*3^(-1 - m)*a^3*(c + d*x)^m*(1 + Cosh[e + f*x])^3*(-(2^m*d*E^ 
(6*e)*(1 + m)*((f*(c + d*x))/d)^m*Gamma[1 + m, (-3*f*(c + d*x))/d]) - 3^(2 
 + m)*d*E^(5*e + (c*f)/d)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, (-2*f*(c + 
d*x))/d] - 5*2^m*3^(2 + m)*d*E^(4*e + (2*c*f)/d)*(1 + m)*((f*(c + d*x))/d) 
^m*Gamma[1 + m, -((f*(c + d*x))/d)] + 5*2^m*3^(2 + m)*d*E^(2*e + (4*c*f)/d 
)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (f*(c + d*x))/d] + 3^(2 + m) 
*d*E^(e + (5*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (2*f*(c + 
 d*x))/d] + 2^m*E^((3*c*f)/d)*(-20*3^(1 + m)*E^(3*e)*f*(c + d*x)*(-((f^2*( 
c + d*x)^2)/d^2))^m + d*E^((3*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma 
[1 + m, (3*f*(c + d*x))/d]))*Sech[(e + f*x)/2]^6)/(d*E^(3*(e + (c*f)/d))*f 
*(1 + m)*(-((f^2*(c + d*x)^2)/d^2))^m))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3799, 3042, 3793, 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.

\(\displaystyle \int (c+d x)^m (a \cosh (e+f x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^m \left (a+a \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^3dx\)

\(\Big \downarrow \) 3799

\(\displaystyle 8 a^3 \int (c+d x)^m \cosh ^6\left (\frac {e}{2}+\frac {f x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 a^3 \int (c+d x)^m \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^6dx\)

\(\Big \downarrow \) 3793

\(\displaystyle 8 a^3 \int \left (\frac {15}{32} \cosh (e+f x) (c+d x)^m+\frac {3}{16} \cosh (2 e+2 f x) (c+d x)^m+\frac {1}{32} \cosh (3 e+3 f x) (c+d x)^m+\frac {5}{16} (c+d x)^m\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 a^3 \left (\frac {3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )}{64 f}+\frac {3\ 2^{-m-6} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{64 f}-\frac {15 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{64 f}-\frac {3\ 2^{-m-6} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )}{64 f}+\frac {5 (c+d x)^{m+1}}{16 d (m+1)}\right )\)

Input:

Int[(c + d*x)^m*(a + a*Cosh[e + f*x])^3,x]
 

Output:

8*a^3*((5*(c + d*x)^(1 + m))/(16*d*(1 + m)) + (3^(-1 - m)*E^(3*e - (3*c*f) 
/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(64*f*(-((f*(c + d*x))/d 
))^m) + (3*2^(-6 - m)*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*( 
c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (15*E^(e - (c*f)/d)*(c + d*x)^m 
*Gamma[1 + m, -((f*(c + d*x))/d)])/(64*f*(-((f*(c + d*x))/d))^m) - (15*E^( 
-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(64*f*((f*(c + d* 
x))/d)^m) - (3*2^(-6 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2 
*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (3^(-1 - m)*E^(-3*e + (3*c*f)/ 
d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + d*x))/d])/(64*f*((f*(c + d*x))/d)^m) 
)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int \left (d x +c \right )^{m} \left (a +a \cosh \left (f x +e \right )\right )^{3}d x\]

Input:

int((d*x+c)^m*(a+a*cosh(f*x+e))^3,x)
 

Output:

int((d*x+c)^m*(a+a*cosh(f*x+e))^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.77 \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^m*(a+a*cosh(f*x+e))^3,x, algorithm="fricas")
 

Output:

-1/24*((a^3*d*m + a^3*d)*cosh((d*m*log(3*f/d) + 3*d*e - 3*c*f)/d)*gamma(m 
+ 1, 3*(d*f*x + c*f)/d) + 9*(a^3*d*m + a^3*d)*cosh((d*m*log(2*f/d) + 2*d*e 
 - 2*c*f)/d)*gamma(m + 1, 2*(d*f*x + c*f)/d) + 45*(a^3*d*m + a^3*d)*cosh(( 
d*m*log(f/d) + d*e - c*f)/d)*gamma(m + 1, (d*f*x + c*f)/d) - 45*(a^3*d*m + 
 a^3*d)*cosh((d*m*log(-f/d) - d*e + c*f)/d)*gamma(m + 1, -(d*f*x + c*f)/d) 
 - 9*(a^3*d*m + a^3*d)*cosh((d*m*log(-2*f/d) - 2*d*e + 2*c*f)/d)*gamma(m + 
 1, -2*(d*f*x + c*f)/d) - (a^3*d*m + a^3*d)*cosh((d*m*log(-3*f/d) - 3*d*e 
+ 3*c*f)/d)*gamma(m + 1, -3*(d*f*x + c*f)/d) - (a^3*d*m + a^3*d)*gamma(m + 
 1, 3*(d*f*x + c*f)/d)*sinh((d*m*log(3*f/d) + 3*d*e - 3*c*f)/d) - 9*(a^3*d 
*m + a^3*d)*gamma(m + 1, 2*(d*f*x + c*f)/d)*sinh((d*m*log(2*f/d) + 2*d*e - 
 2*c*f)/d) - 45*(a^3*d*m + a^3*d)*gamma(m + 1, (d*f*x + c*f)/d)*sinh((d*m* 
log(f/d) + d*e - c*f)/d) + 45*(a^3*d*m + a^3*d)*gamma(m + 1, -(d*f*x + c*f 
)/d)*sinh((d*m*log(-f/d) - d*e + c*f)/d) + 9*(a^3*d*m + a^3*d)*gamma(m + 1 
, -2*(d*f*x + c*f)/d)*sinh((d*m*log(-2*f/d) - 2*d*e + 2*c*f)/d) + (a^3*d*m 
 + a^3*d)*gamma(m + 1, -3*(d*f*x + c*f)/d)*sinh((d*m*log(-3*f/d) - 3*d*e + 
 3*c*f)/d) - 60*(a^3*d*f*x + a^3*c*f)*cosh(m*log(d*x + c)) - 60*(a^3*d*f*x 
 + a^3*c*f)*sinh(m*log(d*x + c)))/(d*f*m + d*f)
 

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x+c)**m*(a+a*cosh(f*x+e))**3,x)
 

Output:

Exception raised: TypeError >> cannot determine truth value of Relational
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.93 \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=-\frac {1}{8} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, e + \frac {3 \, c f}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, e - \frac {3 \, c f}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} - \frac {3}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a^{3} - \frac {3}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} + \frac {{\left (d x + c\right )}^{m + 1} a^{3}}{d {\left (m + 1\right )}} \] Input:

integrate((d*x+c)^m*(a+a*cosh(f*x+e))^3,x, algorithm="maxima")
 

Output:

-1/8*((d*x + c)^(m + 1)*e^(-3*e + 3*c*f/d)*exp_integral_e(-m, 3*(d*x + c)* 
f/d)/d + 3*(d*x + c)^(m + 1)*e^(-e + c*f/d)*exp_integral_e(-m, (d*x + c)*f 
/d)/d + 3*(d*x + c)^(m + 1)*e^(e - c*f/d)*exp_integral_e(-m, -(d*x + c)*f/ 
d)/d + (d*x + c)^(m + 1)*e^(3*e - 3*c*f/d)*exp_integral_e(-m, -3*(d*x + c) 
*f/d)/d)*a^3 - 3/4*((d*x + c)^(m + 1)*e^(-2*e + 2*c*f/d)*exp_integral_e(-m 
, 2*(d*x + c)*f/d)/d + (d*x + c)^(m + 1)*e^(2*e - 2*c*f/d)*exp_integral_e( 
-m, -2*(d*x + c)*f/d)/d - 2*(d*x + c)^(m + 1)/(d*(m + 1)))*a^3 - 3/2*((d*x 
 + c)^(m + 1)*e^(-e + c*f/d)*exp_integral_e(-m, (d*x + c)*f/d)/d + (d*x + 
c)^(m + 1)*e^(e - c*f/d)*exp_integral_e(-m, -(d*x + c)*f/d)/d)*a^3 + (d*x 
+ c)^(m + 1)*a^3/(d*(m + 1))
 

Giac [F]

\[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=\int { {\left (a \cosh \left (f x + e\right ) + a\right )}^{3} {\left (d x + c\right )}^{m} \,d x } \] Input:

integrate((d*x+c)^m*(a+a*cosh(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((a*cosh(f*x + e) + a)^3*(d*x + c)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=\int {\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^m \,d x \] Input:

int((a + a*cosh(e + f*x))^3*(c + d*x)^m,x)
 

Output:

int((a + a*cosh(e + f*x))^3*(c + d*x)^m, x)
 

Reduce [F]

\[ \int (c+d x)^m (a+a \cosh (e+f x))^3 \, dx=\frac {a^{3} \left (e^{6 f x +6 e} \left (d x +c \right )^{m} d m +e^{6 f x +6 e} \left (d x +c \right )^{m} d +9 e^{5 f x +5 e} \left (d x +c \right )^{m} d +45 e^{4 f x +4 e} \left (d x +c \right )^{m} d -45 e^{2 f x +2 e} \left (d x +c \right )^{m} d -9 e^{f x +e} \left (d x +c \right )^{m} d +9 e^{5 f x +5 e} \left (d x +c \right )^{m} d m -\left (d x +c \right )^{m} d -9 e^{3 f x +5 e} \left (\int \frac {e^{2 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m^{2}-9 e^{3 f x +5 e} \left (\int \frac {e^{2 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m -45 e^{3 f x +4 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m^{2}-45 e^{3 f x +4 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m +45 e^{3 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) d^{2} m^{2}+45 e^{3 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) d^{2} m +9 e^{3 f x +e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x} c +e^{2 f x} d x}d x \right ) d^{2} m^{2}+9 e^{3 f x +e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x} c +e^{2 f x} d x}d x \right ) d^{2} m +45 e^{4 f x +4 e} \left (d x +c \right )^{m} d m +60 e^{3 f x +3 e} \left (d x +c \right )^{m} c f -45 e^{2 f x +2 e} \left (d x +c \right )^{m} d m -9 e^{f x +e} \left (d x +c \right )^{m} d m -e^{3 f x +6 e} \left (\int \frac {e^{3 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m^{2}-e^{3 f x +6 e} \left (\int \frac {e^{3 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} m +e^{3 f x +3 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{3 f x +3 e} c +e^{3 f x +3 e} d x}d x \right ) d^{2} m^{2}+e^{3 f x +3 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{3 f x +3 e} c +e^{3 f x +3 e} d x}d x \right ) d^{2} m +60 e^{3 f x +3 e} \left (d x +c \right )^{m} d f x -\left (d x +c \right )^{m} d m \right )}{24 e^{3 f x +3 e} d f \left (m +1\right )} \] Input:

int((d*x+c)^m*(a+a*cosh(f*x+e))^3,x)
 

Output:

(a**3*(e**(6*e + 6*f*x)*(c + d*x)**m*d*m + e**(6*e + 6*f*x)*(c + d*x)**m*d 
 + 9*e**(5*e + 5*f*x)*(c + d*x)**m*d*m + 9*e**(5*e + 5*f*x)*(c + d*x)**m*d 
 + 45*e**(4*e + 4*f*x)*(c + d*x)**m*d*m + 45*e**(4*e + 4*f*x)*(c + d*x)**m 
*d + 60*e**(3*e + 3*f*x)*(c + d*x)**m*c*f + 60*e**(3*e + 3*f*x)*(c + d*x)* 
*m*d*f*x - 45*e**(2*e + 2*f*x)*(c + d*x)**m*d*m - 45*e**(2*e + 2*f*x)*(c + 
 d*x)**m*d - 9*e**(e + f*x)*(c + d*x)**m*d*m - 9*e**(e + f*x)*(c + d*x)**m 
*d - (c + d*x)**m*d*m - (c + d*x)**m*d - e**(6*e + 3*f*x)*int((e**(3*f*x)* 
(c + d*x)**m)/(c + d*x),x)*d**2*m**2 - e**(6*e + 3*f*x)*int((e**(3*f*x)*(c 
 + d*x)**m)/(c + d*x),x)*d**2*m - 9*e**(5*e + 3*f*x)*int((e**(2*f*x)*(c + 
d*x)**m)/(c + d*x),x)*d**2*m**2 - 9*e**(5*e + 3*f*x)*int((e**(2*f*x)*(c + 
d*x)**m)/(c + d*x),x)*d**2*m - 45*e**(4*e + 3*f*x)*int((e**(f*x)*(c + d*x) 
**m)/(c + d*x),x)*d**2*m**2 - 45*e**(4*e + 3*f*x)*int((e**(f*x)*(c + d*x)* 
*m)/(c + d*x),x)*d**2*m + e**(3*e + 3*f*x)*int((c + d*x)**m/(e**(3*e + 3*f 
*x)*c + e**(3*e + 3*f*x)*d*x),x)*d**2*m**2 + e**(3*e + 3*f*x)*int((c + d*x 
)**m/(e**(3*e + 3*f*x)*c + e**(3*e + 3*f*x)*d*x),x)*d**2*m + 45*e**(2*e + 
3*f*x)*int((c + d*x)**m/(e**(f*x)*c + e**(f*x)*d*x),x)*d**2*m**2 + 45*e**( 
2*e + 3*f*x)*int((c + d*x)**m/(e**(f*x)*c + e**(f*x)*d*x),x)*d**2*m + 9*e* 
*(e + 3*f*x)*int((c + d*x)**m/(e**(2*f*x)*c + e**(2*f*x)*d*x),x)*d**2*m**2 
 + 9*e**(e + 3*f*x)*int((c + d*x)**m/(e**(2*f*x)*c + e**(2*f*x)*d*x),x)*d* 
*2*m))/(24*e**(3*e + 3*f*x)*d*f*(m + 1))