3.2.95 \(\int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\) [195]

3.2.95.1 Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\frac {15 b^2 e \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}}{32 d}-\frac {5 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{8 d}+\frac {e (a+b \text {arcsinh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 b^{5/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d} \]

output

1/4*e*(a+b*arcsinh(d*x+c))^(5/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(5 
/2)/d-15/512*b^(5/2)*e*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-15/512*b^(5/2)*e*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)-5/8*b*e*(d*x+c)*(a+b*arcs 
inh(d*x+c))^(3/2)*(1+(d*x+c)^2)^(1/2)/d+15/64*b^2*e*(a+b*arcsinh(d*x+c))^( 
1/2)/d+15/32*b^2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^(1/2)/d
 

3.2.95.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.48 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\frac {e e^{-\frac {2 a}{b}} \left (-b^3 \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+b^3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{32 \sqrt {2} d \sqrt {a+b \text {arcsinh}(c+d x)}} \]

input

Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2),x]
 

output

(e*(-(b^3*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (-2*(a + b*ArcSin 
h[c + d*x]))/b]) + b^3*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, 
 (2*(a + b*ArcSinh[c + d*x]))/b]))/(32*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*Ar 
cSinh[c + d*x]])
 

3.2.95.3 Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6274, 27, 6192, 6227, 6192, 6198, 6234, 3042, 25, 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.

\(\displaystyle \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int e (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \int (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{4} b \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{4} b \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{4} \int \frac {\sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{4} \int -\frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{4} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{4} \int \left (\frac {1}{2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{4} b \left (-\frac {3}{4} b \left (\frac {1}{4} \left (-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arcsinh}(c+d x)}\right )+\frac {1}{2} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {(a+b \text {arcsinh}(c+d x))^{5/2}}{5 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

input

Int[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2),x]
 

output

(e*(((c + d*x)^2*(a + b*ArcSinh[c + d*x])^(5/2))/2 - (5*b*(((c + d*x)*Sqrt 
[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/2 - (a + b*ArcSinh[c + d 
*x])^(5/2)/(5*b) - (3*b*(((c + d*x)^2*Sqrt[a + b*ArcSinh[c + d*x]])/2 + (S 
qrt[a + b*ArcSinh[c + d*x]] - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2] 
*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/4 - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqr 
t[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b)))/4))/4))/4))/ 
d
 

3.2.95.3.1 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

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]))
 

rule 6192

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]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

rule 6234

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]
 

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]
 

3.2.95.4 Maple [F]

\[\int \left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

input

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(5/2),x)
 

output

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(5/2),x)
 

3.2.95.5 Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")
 

output

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

3.2.95.6 Sympy [F]

\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=e \left (\int a^{2} c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b^{2} c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int b^{2} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]

input

integrate((d*e*x+c*e)*(a+b*asinh(d*x+c))**(5/2),x)
 

output

e*(Integral(a**2*c*sqrt(a + b*asinh(c + d*x)), x) + Integral(a**2*d*x*sqrt 
(a + b*asinh(c + d*x)), x) + Integral(b**2*c*sqrt(a + b*asinh(c + d*x))*as 
inh(c + d*x)**2, x) + Integral(2*a*b*c*sqrt(a + b*asinh(c + d*x))*asinh(c 
+ d*x), x) + Integral(b**2*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x)** 
2, x) + Integral(2*a*b*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), x))
 

3.2.95.7 Maxima [F]

\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="maxima")
 

output

integrate((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(5/2), x)
 

3.2.95.8 Giac [F]

\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="giac")
 

output

integrate((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(5/2), x)
 

3.2.95.9 Mupad [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]

input

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^(5/2),x)
 

output

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^(5/2), x)