3.3.5 \(\int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) [205]

3.3.5.1 Optimal result

Integrand size = 25, antiderivative size = 217 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \]

output

1/16*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)/d/b^(1/2)-1/16*e^3*erfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1 
/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)/b^(1/2)-1/32*e^3*exp(4*a/b)*erf(2*(a+b* 
arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/b^(1/2)+1/32*e^3*erfi(2*(a+b*arc 
sinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(4*a/b)/b^(1/2)
 

3.3.5.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-2 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \left (-2 \sqrt {2} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )}{32 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

input

Integrate[(c*e + d*e*x)^3/Sqrt[a + b*ArcSinh[c + d*x]],x]
 

output

(e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c 
+ d*x]))/b] - 2*Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Ga 
mma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[a/b + ArcSinh 
[c + d*x]]*(-2*Sqrt[2]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2* 
a)/b)*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(32*d*E^((4*a)/b)*Sqrt 
[a + b*ArcSinh[c + d*x]])
 

3.3.5.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6274, 27, 6195, 25, 5971, 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.

\(\displaystyle \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {e^3 \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {e^3 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {e^3 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))}{b d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b d}\)

input

Int[(c*e + d*e*x)^3/Sqrt[a + b*ArcSinh[c + d*x]],x]
 

output

(e^3*(-1/32*(Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d* 
x]])/Sqrt[b]]) + (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A 
rcSinh[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSi 
nh[c + d*x]])/Sqrt[b]])/(32*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[ 
2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*E^((2*a)/b))))/(b*d)
 

3.3.5.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6195

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
1/(b*c^(m + 1))   Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, 
 a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 6274

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]
 

3.3.5.4 Maple [F]

\[\int \frac {\left (d e x +c e \right )^{3}}{\sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]

input

int((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x)
 

output

int((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x)
 

3.3.5.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.3.5.6 Sympy [F]

\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=e^{3} \left (\int \frac {c^{3}}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx + \int \frac {d^{3} x^{3}}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx + \int \frac {3 c d^{2} x^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx + \int \frac {3 c^{2} d x}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx\right ) \]

input

integrate((d*e*x+c*e)**3/(a+b*asinh(d*x+c))**(1/2),x)
 

output

e**3*(Integral(c**3/sqrt(a + b*asinh(c + d*x)), x) + Integral(d**3*x**3/sq 
rt(a + b*asinh(c + d*x)), x) + Integral(3*c*d**2*x**2/sqrt(a + b*asinh(c + 
 d*x)), x) + Integral(3*c**2*d*x/sqrt(a + b*asinh(c + d*x)), x))
 

3.3.5.7 Maxima [F]

\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]

input

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")
 

output

integrate((d*e*x + c*e)^3/sqrt(b*arcsinh(d*x + c) + a), x)
 

3.3.5.8 Giac [F]

\[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]

input

integrate((d*e*x+c*e)^3/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")
 

output

integrate((d*e*x + c*e)^3/sqrt(b*arcsinh(d*x + c) + a), x)
 

3.3.5.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^3}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \]

input

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(1/2),x)
 

output

int((c*e + d*e*x)^3/(a + b*asinh(c + d*x))^(1/2), x)