Integrand size = 25, antiderivative size = 262 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{b d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d}+\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d} \]
-1/4*e^3*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)/b^(3/2)/d-1/4*e^3*erfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/ 2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d/exp(2*a/b)+1/4*e^3*exp(4*a/b)*erf(2*(a+b*ar csinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/d+1/4*e^3*erfi(2*(a+b*arcsin h(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/d/exp(4*a/b)-2*e^3*(d*x+c)^3*(1+ (d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(1/2)
Time = 0.34 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.97 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-e^{\frac {4 a}{b}} \left (-\sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-2 \sinh (2 \text {arcsinh}(c+d x))+\sinh (4 \text {arcsinh}(c+d x))\right )\right )}{4 b d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
(e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamm a[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] - E^((4*a)/b)*(-(Sqrt[2]*E^((2*a)/ b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] ) + E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (4*(a + b*ArcSinh[ c + d*x]))/b] - 2*Sinh[2*ArcSinh[c + d*x]] + Sinh[4*ArcSinh[c + d*x]])))/( 4*b*d*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])
Time = 0.58 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6274, 27, 6193, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x)^3 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{d}\) |
(e^3*((-2*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*Sqrt[a + b*ArcSinh[c + d*x ]]) + (2*((Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x] ])/Sqrt[b]])/8 - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A rcSinh[c + d*x]])/Sqrt[b]])/4 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSi nh[c + d*x]])/Sqrt[b]])/(8*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2 ]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b))))/b^2))/d
3.3.11.3.1 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_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
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]
\[\int \frac {\left (d e x +c e \right )^{3}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a*sqrt(a + b*asinh(c + d*x)) + b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)), x) + Integral(d**3*x**3/(a*sqrt(a + b*asinh(c + d* x)) + b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)), x) + Integral(3*c*d**2 *x**2/(a*sqrt(a + b*asinh(c + d*x)) + b*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)), x) + Integral(3*c**2*d*x/(a*sqrt(a + b*asinh(c + d*x)) + b*sqrt( a + b*asinh(c + d*x))*asinh(c + d*x)), x))
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]