\(\int (d+e x)^m (a+b \text {arcsinh}(c x)) \, dx\) [31]

Optimal result

Integrand size = 16, antiderivative size = 179 \[ \int (d+e x)^m (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c (d+e x)^{2+m} \sqrt {1-\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}}} \sqrt {1-\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \operatorname {AppellF1}\left (2+m,\frac {1}{2},\frac {1}{2},3+m,\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}},\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}\right )}{e^2 (1+m) (2+m) \sqrt {1+c^2 x^2}}+\frac {(d+e x)^{1+m} (a+b \text {arcsinh}(c x))}{e (1+m)} \] Output:

-b*c*(e*x+d)^(2+m)*(1-(e*x+d)/(d-e/(-c^2)^(1/2)))^(1/2)*(1-(e*x+d)/(d+e/(- 
c^2)^(1/2)))^(1/2)*AppellF1(2+m,1/2,1/2,3+m,(e*x+d)/(d-e/(-c^2)^(1/2)),(e* 
x+d)/(d+e/(-c^2)^(1/2)))/e^2/(1+m)/(2+m)/(c^2*x^2+1)^(1/2)+(e*x+d)^(1+m)*( 
a+b*arcsinh(c*x))/e/(1+m)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6243, 514, 150}

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.

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

\(\Big \downarrow \) 6243

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arcsinh}(c x))}{e (m+1)}-\frac {b c \int \frac {(d+e x)^{m+1}}{\sqrt {c^2 x^2+1}}dx}{e (m+1)}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arcsinh}(c x))}{e (m+1)}-\frac {b c \sqrt {1-\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}}} \sqrt {1-\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} \int \frac {(d+e x)^{m+1}}{\sqrt {1-\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}}} \sqrt {1-\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}d(d+e x)}{e^2 (m+1) \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(d+e x)^{m+1} (a+b \text {arcsinh}(c x))}{e (m+1)}-\frac {b c \sqrt {1-\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}}} \sqrt {1-\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} (d+e x)^{m+2} \operatorname {AppellF1}\left (m+2,\frac {1}{2},\frac {1}{2},m+3,\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}},\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}\right )}{e^2 (m+1) (m+2) \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-((b*c*(d + e*x)^(2 + m)*Sqrt[1 - (d + e*x)/(d - e/Sqrt[-c^2])]*Sqrt[1 - ( 
d + e*x)/(d + e/Sqrt[-c^2])]*AppellF1[2 + m, 1/2, 1/2, 3 + m, (d + e*x)/(d 
 - e/Sqrt[-c^2]), (d + e*x)/(d + e/Sqrt[-c^2])])/(e^2*(1 + m)*(2 + m)*Sqrt 
[1 + c^2*x^2])) + ((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x]))/(e*(1 + m))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 6243

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] 
 - Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( 
n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n 
, 0] && NeQ[m, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

b*((e*x + d)*(e*x + d)^m*log(c*x + sqrt(c^2*x^2 + 1))/(e*(m + 1)) - integr 
ate((c^2*e*x^2 + c^2*d*x)*(e*x + d)^m/(c^2*e*(m + 1)*x^2 + e*(m + 1)), x) 
- integrate((c*e*x + c*d)*(e*x + d)^m/(c^3*e*(m + 1)*x^3 + c*e*(m + 1)*x + 
 (c^2*e*(m + 1)*x^2 + e*(m + 1))*sqrt(c^2*x^2 + 1)), x)) + (e*x + d)^(m + 
1)*a/(e*(m + 1))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^m (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (e x +d \right )^{m} a d +\left (e x +d \right )^{m} a e x +\left (\int \left (e x +d \right )^{m} \mathit {asinh} \left (c x \right )d x \right ) b e m +\left (\int \left (e x +d \right )^{m} \mathit {asinh} \left (c x \right )d x \right ) b e}{e \left (m +1\right )} \] Input:

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

Output:

((d + e*x)**m*a*d + (d + e*x)**m*a*e*x + int((d + e*x)**m*asinh(c*x),x)*b* 
e*m + int((d + e*x)**m*asinh(c*x),x)*b*e)/(e*(m + 1))