3.3.25 \(\int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\) [225]

3.3.25.1 Optimal result

Integrand size = 23, antiderivative size = 252 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {4 e}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {32 e (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {8 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d} \]

output

-4/15*e/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-8/15*e*(d*x+c)^2/b^2/d/(a+b*arcsi 
nh(d*x+c))^(3/2)+8/15*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)/b^(7/2)/d+8/15*e*erfi(2^(1/2)*(a+b*arcsinh(d*x+c 
))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(2*a/b)-2/5*e*(d*x+c)*(1+( 
d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^(5/2)-32/15*e*(d*x+c)*(1+(d*x+c)^ 
2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))^(1/2)
 

3.3.25.2 Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.93 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {e \left ((a+b \text {arcsinh}(c+d x)) \left (e^{-\frac {2 a}{b}} \left (2 e^{2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (4 a+b+4 b \text {arcsinh}(c+d x))+8 \sqrt {2} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )+e^{-2 \text {arcsinh}(c+d x)} \left (-8 a+2 b-8 b \text {arcsinh}(c+d x)+8 \sqrt {2} e^{2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )+3 b^2 \sinh (2 \text {arcsinh}(c+d x))\right )}{15 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]

input

Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(7/2),x]
 

output

-1/15*(e*((a + b*ArcSinh[c + d*x])*((2*E^(2*(a/b + ArcSinh[c + d*x]))*(4*a 
 + b + 4*b*ArcSinh[c + d*x]) + 8*Sqrt[2]*b*(-((a + b*ArcSinh[c + d*x])/b)) 
^(3/2)*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b])/E^((2*a)/b) + (-8*a + 
2*b - 8*b*ArcSinh[c + d*x] + 8*Sqrt[2]*E^(2*(a/b + ArcSinh[c + d*x]))*Sqrt 
[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (2*(a + b*Arc 
Sinh[c + d*x]))/b])/E^(2*ArcSinh[c + d*x])) + 3*b^2*Sinh[2*ArcSinh[c + d*x 
]]))/(b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))
 

3.3.25.3 Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6274, 27, 6194, 6198, 6233, 6193, 3042, 3788, 26, 2611, 2633, 2634}

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.

\(\displaystyle \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int \frac {c+d x}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6194

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

\(\Big \downarrow \) 6198

\(\displaystyle \frac {e \left (\frac {4 \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 6233

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \int \frac {c+d x}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 6193

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (\frac {4 \left (-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {4 \left (-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {e \left (\frac {4 \left (-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {4 \left (-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {i e^{-\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{3 b}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \int \frac {e^{\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \left (\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\int e^{\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \left (\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}\right )}{d}\)

input

Int[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(7/2),x]
 

output

(e*((-2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(5*b*(a + b*ArcSinh[c + d*x])^(5/ 
2)) - 4/(15*b^2*(a + b*ArcSinh[c + d*x])^(3/2)) + (4*((-2*(c + d*x)^2)/(3* 
b*(a + b*ArcSinh[c + d*x])^(3/2)) + (4*((-2*(c + d*x)*Sqrt[1 + (c + d*x)^2 
])/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + (2*((Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]* 
Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi/ 
2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*E^((2*a)/b)))) 
/b^2))/(3*b)))/(5*b)))/d
 

3.3.25.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 6193

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]
 

rule 6194

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] + (- 
Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ 
Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*A 
rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && 
IGtQ[m, 0] && LtQ[n, -2]
 

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 6233

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

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]
 

3.3.25.4 Maple [F]

\[\int \frac {d e x +c e}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]

input

int((d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x)
 

output

int((d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x)
 

3.3.25.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="fricas")
 

output

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

3.3.25.6 Sympy [F]

\[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=e \left (\int \frac {c}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

input

integrate((d*e*x+c*e)/(a+b*asinh(d*x+c))**(7/2),x)
 

output

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

3.3.25.7 Maxima [F]

\[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

input

integrate((d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="maxima")
 

output

integrate((d*e*x + c*e)/(b*arcsinh(d*x + c) + a)^(7/2), x)
 

3.3.25.8 Giac [F]

\[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

input

integrate((d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="giac")
 

output

integrate((d*e*x + c*e)/(b*arcsinh(d*x + c) + a)^(7/2), x)
 

3.3.25.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]

input

int((c*e + d*e*x)/(a + b*asinh(c + d*x))^(7/2),x)
 

output

int((c*e + d*e*x)/(a + b*asinh(c + d*x))^(7/2), x)