Integrand size = 16, antiderivative size = 389 \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{64 d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {15 b^{5/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 b^{5/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \]
-c*(d*x+c)*(a+b*arcsinh(d*x+c))^(5/2)/d^2+1/4*(a+b*arcsinh(d*x+c))^(5/2)*c osh(2*arcsinh(d*x+c))/d^2-5/16*b*(a+b*arcsinh(d*x+c))^(3/2)*sinh(2*arcsinh (d*x+c))/d^2-15/512*b^(5/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1 /2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2-15/512*b^(5/2)*erfi(2^(1/2)*(a+b*arcsinh (d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2/exp(2*a/b)-15/16*b^(5/2)*c*ex p(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2+15/16*b^(5/2)* c*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2/exp(a/b)+5/2*b*c*( a+b*arcsinh(d*x+c))^(3/2)*(1+(d*x+c)^2)^(1/2)/d^2-15/4*b^2*c*(d*x+c)*(a+b* arcsinh(d*x+c))^(1/2)/d^2+15/64*b^2*cosh(2*arcsinh(d*x+c))*(a+b*arcsinh(d* x+c))^(1/2)/d^2
Leaf count is larger than twice the leaf count of optimal. \(939\) vs. \(2(389)=778\).
Time = 8.10 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.41 \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\frac {-1920 b^2 c^2 \sqrt {a+b \text {arcsinh}(c+d x)}-1920 b^2 c d x \sqrt {a+b \text {arcsinh}(c+d x)}+1280 a b c \sqrt {1+c^2+2 c d x+d^2 x^2} \sqrt {a+b \text {arcsinh}(c+d x)}-1024 a b c^2 \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-1024 a b c d x \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+1280 b^2 c \sqrt {1+c^2+2 c d x+d^2 x^2} \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-512 b^2 c^2 \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-512 b^2 c d x \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}+128 a^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+120 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+256 a b \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+128 b^2 \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))-128 a^2 \sqrt {b} c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+480 b^{5/2} c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-15 b^{5/2} \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {256 a^2 b c e^{a/b} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {256 a^2 b c e^{-\frac {a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}+128 a^2 \sqrt {b} c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-480 b^{5/2} c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+32 \sqrt {b} \left (4 a^2-15 b^2\right ) c \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 )+15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-160 a b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))-160 b^2 \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{512 d^2} \]
(-1920*b^2*c^2*Sqrt[a + b*ArcSinh[c + d*x]] - 1920*b^2*c*d*x*Sqrt[a + b*Ar cSinh[c + d*x]] + 1280*a*b*c*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]*Sqrt[a + b* ArcSinh[c + d*x]] - 1024*a*b*c^2*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d *x]] - 1024*a*b*c*d*x*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] + 1280 *b^2*c*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]*ArcSinh[c + d*x]*Sqrt[a + b*ArcSi nh[c + d*x]] - 512*b^2*c^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]] - 512*b^2*c*d*x*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]] + 128*a^2 *Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 120*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 256*a*b*ArcSinh[c + d*x]*Sq rt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 128*b^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] - 128*a^2*Sqr t[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + 480 *b^(5/2)*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 15*b^(5/2)*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + (256*a^2*b*c*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[ 3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a + b*ArcSinh[c + d*x]] + (256*a^2*b*c* Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/ b)])/(E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]]) + 128*a^2*Sqrt[b]*c*Sqrt[Pi]*E rfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 480*b^(5/2)*c*Sqrt[P i]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 32*Sqrt[b]*(4...
Time = 1.36 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6274, 25, 27, 6245, 7267, 7292, 7293, 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.
| \(\displaystyle \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int x (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -d x (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 6245 |
| \(\displaystyle -\frac {\int -d x \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}d\text {arcsinh}(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -\frac {2 \int -d x \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^2}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \int -d x (a+b \text {arcsinh}(c+d x))^3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \int \left (c \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) (a+b \text {arcsinh}(c+d x))^3+\frac {1}{2} \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) (a+b \text {arcsinh}(c+d x))^3\right )d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\frac {15}{32} \sqrt {\pi } b^{7/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {15}{512} \sqrt {\frac {\pi }{2}} b^{7/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {15}{32} \sqrt {\pi } b^{7/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {15}{512} \sqrt {\frac {\pi }{2}} b^{7/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {15}{8} b^3 c \sqrt {a+b \text {arcsinh}(c+d x)} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {15}{128} b^3 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {5}{32} b^2 (a+b \text {arcsinh}(c+d x))^{3/2} \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {5}{4} b^2 c (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{2} b c (a+b \text {arcsinh}(c+d x))^{5/2} \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{8} b (a+b \text {arcsinh}(c+d x))^{5/2} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b d^2}\) |
(-2*((-15*b^3*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[(2*a)/b - (2*(a + b*ArcSin h[c + d*x]))/b])/128 - (b*(a + b*ArcSinh[c + d*x])^(5/2)*Cosh[(2*a)/b - (2 *(a + b*ArcSinh[c + d*x]))/b])/8 - (5*b^2*c*(a + b*ArcSinh[c + d*x])^(3/2) *Cosh[a/b - (a + b*ArcSinh[c + d*x])/b])/4 + (15*b^(7/2)*c*E^(a/b)*Sqrt[Pi ]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/32 + (15*b^(7/2)*E^((2*a)/b)* Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/512 - (15* b^(7/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*E^(a/b) ) + (15*b^(7/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqr t[b]])/(512*E^((2*a)/b)) - (5*b^2*(a + b*ArcSinh[c + d*x])^(3/2)*Sinh[(2*a )/b - (2*(a + b*ArcSinh[c + d*x]))/b])/32 - (15*b^3*c*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[a/b - (a + b*ArcSinh[c + d*x])/b])/8 - (b*c*(a + b*ArcSinh[c + d*x])^(5/2)*Sinh[a/b - (a + b*ArcSinh[c + d*x])/b])/2))/(b*d^2)
3.2.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[1/c^(m + 1) Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[ x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int x \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
Timed out. \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]