\(\int (e+f x)^m (a+b \text {arctanh}(c+d x)) \, dx\) [52]

Optimal result

Integrand size = 18, antiderivative size = 162 \[ \int (e+f x)^m (a+b \text {arctanh}(c+d x)) \, dx=\frac {(e+f x)^{1+m} (a+b \text {arctanh}(c+d x))}{f (1+m)}+\frac {b d (e+f x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {d (e+f x)}{d e-f-c f}\right )}{2 f (d e-(1+c) f) (1+m) (2+m)}-\frac {b d (e+f x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {d (e+f x)}{d e+f-c f}\right )}{2 f (d e+f-c f) (1+m) (2+m)} \] Output:

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

Mathematica [F]

\[ \int (e+f x)^m (a+b \text {arctanh}(c+d x)) \, dx=\int (e+f x)^m (a+b \text {arctanh}(c+d x)) \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.37, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6661, 6478, 485, 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.

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

\(\Big \downarrow \) 6661

\(\displaystyle \frac {\int \left (e-\frac {c f}{d}+\frac {f (c+d x)}{d}\right )^m (a+b \text {arctanh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 6478

\(\displaystyle \frac {\frac {d (a+b \text {arctanh}(c+d x)) \left (\frac {f (c+d x)}{d}-\frac {c f}{d}+e\right )^{m+1}}{f (m+1)}-\frac {b d \int \frac {\left (e-\frac {c f}{d}+\frac {f (c+d x)}{d}\right )^{m+1}}{1-(c+d x)^2}d(c+d x)}{f (m+1)}}{d}\)

\(\Big \downarrow \) 485

\(\displaystyle \frac {\frac {d (a+b \text {arctanh}(c+d x)) \left (\frac {f (c+d x)}{d}-\frac {c f}{d}+e\right )^{m+1}}{f (m+1)}-\frac {b d \int \left (\frac {\left (e-\frac {c f}{d}+\frac {f (c+d x)}{d}\right )^{m+1}}{2 (-c-d x+1)}+\frac {\left (e-\frac {c f}{d}+\frac {f (c+d x)}{d}\right )^{m+1}}{2 (c+d x+1)}\right )d(c+d x)}{f (m+1)}}{d}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d*(e - (c*f)/d + (f*(c + d*x))/d)^(1 + m)*(a + b*ArcTanh[c + d*x]))/(f*( 
1 + m)) - (b*d*(-1/2*(d*(e - (c*f)/d + (f*(c + d*x))/d)^(2 + m)*Hypergeome 
tric2F1[1, 2 + m, 3 + m, (d*e - c*f + f*(c + d*x))/(d*e - f - c*f)])/((d*e 
 - (1 + c)*f)*(2 + m)) + (d*(e - (c*f)/d + (f*(c + d*x))/d)^(2 + m)*Hyperg 
eometric2F1[1, 2 + m, 3 + m, (d*e - c*f + f*(c + d*x))/(d*e + f - c*f)])/( 
2*(d*e + f - c*f)*(2 + m))))/(f*(1 + m)))/d
 

Defintions of rubi rules used

rule 485

Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand 
Integrand[(c + d*x)^n, 1/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, n}, x] & 
&  !IntegerQ[2*n]
 

rule 2009

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

rule 6478

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])/(e*(q + 1))), x] - Simp[b 
*(c/(e*(q + 1)))   Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, 
 b, c, d, e, q}, x] && NeQ[q, -1]
 

rule 6661

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG 
tQ[p, 0]
 

Maple [F]

\[\int \left (f x +e \right )^{m} \left (a +b \,\operatorname {arctanh}\left (d x +c \right )\right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arctanh(d*x + c) + a)*(f*x + e)^m, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((e + f*x)**m*atanh(c + d*x)*b*c*e*m + (e + f*x)**m*atanh(c + d*x)*b*c*f*m 
*x + (e + f*x)**m*a*c*e*m + (e + f*x)**m*a*c*f*m*x + (e + f*x)**m*b*e - in 
t((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x + 
2*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 + 
d**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*c**2*e*f*m**2 - 
int((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x 
+ 2*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 
+ d**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*c**2*e*f*m + i 
nt((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x + 
 2*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 + 
 d**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*c*d*e**2*m**2 + 
 int((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x 
 + 2*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 
 + d**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*c*d*e**2*m + 
int((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x 
+ 2*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 
+ d**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*e*f*m**2 + int 
((e + f*x)**m/(c**2*e*m + c**2*e + c**2*f*m*x + c**2*f*x + 2*c*d*e*m*x + 2 
*c*d*e*x + 2*c*d*f*m*x**2 + 2*c*d*f*x**2 + d**2*e*m*x**2 + d**2*e*x**2 + d 
**2*f*m*x**3 + d**2*f*x**3 - e*m - e - f*m*x - f*x),x)*b*e*f*m + int(((...