Integrand size = 14, antiderivative size = 216 \[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}}{d}+\frac {105 b^{7/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {105 b^{7/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d} \]
35/4*b^2*(d*x+c)*(a+b*arcsinh(d*x+c))^(3/2)/d+(d*x+c)*(a+b*arcsinh(d*x+c)) ^(7/2)/d+105/32*b^(7/2)*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*P i^(1/2)/d+105/32*b^(7/2)*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2) /d/exp(a/b)-7/2*b*(a+b*arcsinh(d*x+c))^(5/2)*(1+(d*x+c)^2)^(1/2)/d-105/8*b ^3*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(698\) vs. \(2(216)=432\).
Time = 3.27 (sec) , antiderivative size = 698, normalized size of antiderivative = 3.23 \[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\frac {16 a^3 e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )+12 a^2 \sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-3 \sqrt {1+(c+d x)^2}+2 (c+d x) \text {arcsinh}(c+d x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+6 a \sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (2 \sqrt {1+(c+d x)^2} (a-5 b \text {arcsinh}(c+d x))+b (c+d x) \left (15+4 \text {arcsinh}(c+d x)^2\right )\right )+\left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c+d x)} \left (2 b (c+d x) \left (-10 a+35 b \text {arcsinh}(c+d x)+4 b \text {arcsinh}(c+d x)^3\right )+\sqrt {1+(c+d x)^2} \left (-4 a^2+4 a b \text {arcsinh}(c+d x)-7 b^2 \left (15+4 \text {arcsinh}(c+d x)^2\right )\right )\right )+\left (8 a^3+36 a^2 b+90 a b^2+105 b^3\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+\left (-8 a^3+36 a^2 b-90 a b^2+105 b^3\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{32 d} \]
((16*a^3*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + Arc Sinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSin h[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + 12*a^2*Sqr t[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2 *(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSin h[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[S qrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 6*a*Sqrt[b ]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(2*Sqrt[1 + (c + d*x)^2]*(a - 5* b*ArcSinh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcSinh[c + d*x]^2)) + (4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(-Cos h[a/b] + Sinh[a/b]) + (4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*Ar cSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(4*Sqrt[b]*Sqr t[a + b*ArcSinh[c + d*x]]*(2*b*(c + d*x)*(-10*a + 35*b*ArcSinh[c + d*x] + 4*b*ArcSinh[c + d*x]^3) + Sqrt[1 + (c + d*x)^2]*(-4*a^2 + 4*a*b*ArcSinh[c + d*x] - 7*b^2*(15 + 4*ArcSinh[c + d*x]^2))) + (8*a^3 + 36*a^2*b + 90*a*b^ 2 + 105*b^3)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b ] - Sinh[a/b]) + (-8*a^3 + 36*a^2*b - 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erf[Sqr t[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(32*d)
Time = 1.03 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6273, 6187, 6213, 6187, 6213, 6189, 3042, 3788, 26, 2611, 2633, 2634}
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.
| \(\displaystyle \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 6273 |
| \(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \int (a+b \text {arcsinh}(c+d x))^{3/2}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \int \frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} b \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6189 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \left (\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcSinh[c + d*x])^(7/2) - (7*b*(Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2) - (5*b*((c + d*x)*(a + b*ArcSinh[c + d*x])^(3 /2) - (3*b*(Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]] + (-1/2*(Sq rt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]) - (Sqrt[ b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*E^(a/b)))/2))/2 ))/2))/2)/d
3.3.2.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]
Exception generated. \[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
\[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^{7/2} \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]