Integrand size = 25, antiderivative size = 361 \[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^4 (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}}{5 d}+\frac {\sqrt {b} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {\sqrt {b} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{320 d}-\frac {\sqrt {b} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {\sqrt {b} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {\sqrt {b} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{320 d} \]
1/1600*e^4*exp(5*a/b)*erf(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1 /2)*5^(1/2)*Pi^(1/2)/d-1/1600*e^4*erfi(5^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/ b^(1/2))*b^(1/2)*5^(1/2)*Pi^(1/2)/d/exp(5*a/b)-1/192*e^4*exp(3*a/b)*erf(3^ (1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/d+1/192 *e^4*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^( 1/2)/d/exp(3*a/b)+1/32*e^4*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2) )*b^(1/2)*Pi^(1/2)/d-1/32*e^4*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^( 1/2)*Pi^(1/2)/d/exp(a/b)+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^(1/2)/d
Time = 0.52 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.95 \[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^4 e^{-\frac {5 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-150 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+3 \sqrt {5} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )-25 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+150 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+25 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-3 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{2400 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
(e^4*Sqrt[a + b*ArcSinh[c + d*x]]*(-150*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[ c + d*x])/b)]*Gamma[3/2, a/b + ArcSinh[c + d*x]] + 3*Sqrt[5]*Sqrt[a/b + Ar cSinh[c + d*x]]*Gamma[3/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 25*Sqrt[3]*E ^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 150*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)] + 25*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c + d*x]))/b] - 3*Sqrt[5]*E^((10* a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(2400*d*E^((5*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6274, 27, 6192, 6234, 25, 3042, 26, 3793, 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.
| \(\displaystyle \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^4 (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int (c+d x)^4 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} \int -\frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{10} \int \frac {\sinh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{10} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^5}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{10} i \left (\frac {5}{16} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{32} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} i \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{16} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {5}{32} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} i \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )}{d}\) |
(e^4*(((c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/5 - (I/10)*(((5*I)/16)*Sq rt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - ((5*I)/ 32)*Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x ]])/Sqrt[b]] + (I/32)*Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - (((5*I)/16)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b) + (((5*I)/32)*Sqrt[b]*Sqrt[Pi/3]*E rfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/E^((3*a)/b) - ((I/32) *Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/ E^((5*a)/b))))/d
3.2.80.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && 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]
\[\int \left (d e x +c e \right )^{4} \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}d x\]
Exception generated. \[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=e^{4} \left (\int c^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int d^{4} x^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 4 c d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 6 c^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 4 c^{3} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4*sqrt(a + b*asinh(c + d*x)), x) + Integral(d**4*x**4*sq rt(a + b*asinh(c + d*x)), x) + Integral(4*c*d**3*x**3*sqrt(a + b*asinh(c + d*x)), x) + Integral(6*c**2*d**2*x**2*sqrt(a + b*asinh(c + d*x)), x) + In tegral(4*c**3*d*x*sqrt(a + b*asinh(c + d*x)), x))
\[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{4} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
\[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{4} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int (c e+d e x)^4 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]