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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5828, 774, 138} \[ \int (d+e x)^m (a+b \text {arcsinh}(c x)) \, dx=\frac {(d+e x)^{m+1} (a+b \text {arcsinh}(c x))}{e (m+1)}-\frac {b c \sqrt {1-\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}}} \sqrt {1-\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} (d+e x)^{m+2} \operatorname {AppellF1}\left (m+2,\frac {1}{2},\frac {1}{2},m+3,\frac {d+e x}{d-\frac {e}{\sqrt {-c^2}}},\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}\right )}{e^2 (m+1) (m+2) \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

-((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])]*AppellF
1[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)*Sq
rt[1 + c^2*x^2])) + ((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x]))/(e*(1 + m))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> 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] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 774

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[(a + c*x^
2)^p/(e*(1 - (d + e*x)/(d + e*(q/c)))^p*(1 - (d + e*x)/(d - e*(q/c)))^p), Subst[Int[x^m*Simp[1 - x/(d + e*(q/c
)), x]^p*Simp[1 - x/(d - e*(q/c)), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a
*e^2, 0] &&  !IntegerQ[p]

Rule 5828

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] - Dist[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]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

b*((e*x + d)*(e*x + d)^m*log(c*x + sqrt(c^2*x^2 + 1))/(e*(m + 1)) - integrate((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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c*x))*(d + e*x)^m,x)

[Out]

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