\(\int (e+f x)^m (a+b \coth ^{-1}(c+d x)) \, dx\) [119]

Optimal result

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

[Out]

(f*x+e)^(1+m)*(a+b*arccoth(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)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6247, 6064, 726, 70} \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^{m+1} \left (a+b \coth ^{-1}(c+d x)\right )}{f (m+1)}+\frac {b d (e+f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {d (e+f x)}{d e-c f-f}\right )}{2 f (m+1) (m+2) (d e-(c+1) f)}-\frac {b d (e+f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {d (e+f x)}{d e-c f+f}\right )}{2 f (m+1) (m+2) (-c f+d e+f)} \]

[In]

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

[Out]

((e + f*x)^(1 + m)*(a + b*ArcCoth[c + d*x]))/(f*(1 + m)) + (b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m,
3 + m, (d*(e + f*x))/(d*e - f - c*f)])/(2*f*(d*e - (1 + c)*f)*(1 + m)*(2 + m)) - (b*d*(e + f*x)^(2 + m)*Hyperg
eometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + f - c*f)])/(2*f*(d*e + f - c*f)*(1 + m)*(2 + m))

Rule 70

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

Rule 726

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 6064

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b
*ArcCoth[c*x])/(e*(q + 1))), x] - Dist[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 6247

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^m \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^{1+m} \left (a+b \coth ^{-1}(c+d x)\right )}{f (1+m)}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{1-x^2} \, dx,x,c+d x\right )}{f (1+m)} \\ & = \frac {(e+f x)^{1+m} \left (a+b \coth ^{-1}(c+d x)\right )}{f (1+m)}-\frac {b \text {Subst}\left (\int \left (\frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{2 (1-x)}+\frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{2 (1+x)}\right ) \, dx,x,c+d x\right )}{f (1+m)} \\ & = \frac {(e+f x)^{1+m} \left (a+b \coth ^{-1}(c+d x)\right )}{f (1+m)}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{1-x} \, dx,x,c+d x\right )}{2 f (1+m)}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{1+x} \, dx,x,c+d x\right )}{2 f (1+m)} \\ & = \frac {(e+f x)^{1+m} \left (a+b \coth ^{-1}(c+d x)\right )}{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)} \\ \end{align*}

Mathematica [F]

\[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right ) \left (e + f x\right )^{m}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

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]

\[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\int {\left (e+f\,x\right )}^m\,\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right ) \,d x \]

[In]

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

[Out]

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