\(\int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx\) [194]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Int}\left (x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)),x\right ) \] Output:

Defer(Int)(x*(c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x)),x)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(26)=52\).

Time = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \left (\sqrt [3]{1+c^2 x^2} \left (8 a+8 a c^2 x^2-3 b c x \sqrt {1+c^2 x^2}+8 \left (b+b c^2 x^2\right ) \text {arcsinh}(c x)\right )-5 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-c^2 x^2\right )\right )}{64 c^2} \] Input:

Integrate[x*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]),x]
 

Output:

(3*((1 + c^2*x^2)^(1/3)*(8*a + 8*a*c^2*x^2 - 3*b*c*x*Sqrt[1 + c^2*x^2] + 8 
*(b + b*c^2*x^2)*ArcSinh[c*x]) - 5*b*c*x*Hypergeometric2F1[1/6, 1/2, 3/2, 
-(c^2*x^2)]))/(64*c^2)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(836\) vs. \(2(26)=52\).

Time = 0.85 (sec) , antiderivative size = 836, normalized size of antiderivative = 36.35, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6213, 211, 235, 214, 233, 833, 760, 2418}

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.

\(\displaystyle \int x \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \int \left (c^2 x^2+1\right )^{5/6}dx}{8 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \int \frac {1}{\sqrt [6]{c^2 x^2+1}}dx+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 235

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \int \frac {1}{\left (c^2 x^2+1\right )^{7/6}}dx\right )+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 214

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt [3]{c^2 x^2+1} \int \frac {1}{\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}d\frac {x}{\sqrt {c^2 x^2+1}}\right )+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \int \frac {\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {5}{8} \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{8} x \left (c^2 x^2+1\right )^{5/6}\right )}{8 c}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 \left (c^2 x^2+1\right )^{4/3} (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {3 b \left (\frac {3}{8} \left (c^2 x^2+1\right )^{5/6} x+\frac {5}{8} \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )\right )}{8 c}\)

Input:

Int[x*(1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]),x]
 

Output:

(3*(1 + c^2*x^2)^(4/3)*(a + b*ArcSinh[c*x]))/(8*c^2) - (3*b*((3*x*(1 + c^2 
*x^2)^(5/6))/8 + (5*((3*x)/(2*(1 + c^2*x^2)^(1/6)) + (3*((1 + c^2*x^2)^(-1 
))^(1/3)*Sqrt[-((c^2*x^2)/(1 + c^2*x^2))]*(1 + c^2*x^2)^(5/6)*((-2*Sqrt[-1 
 + x^3/(1 + c^2*x^2)^(3/2)])/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^ 
(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/ 
3))*Sqrt[(1 + x^2/(1 + c^2*x^2) + (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 
- Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))^2]*EllipticE[ArcSin[(1 + 
Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^ 
2)/(1 + c^2*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-1 + x^3/(1 + c^2*x^2)^( 
3/2)]*Sqrt[-((1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - 
 (c^2*x^2)/(1 + c^2*x^2))^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3]) 
*(1 - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))*Sqrt[(1 + x^2/(1 + c^2*x^2) + ( 
1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2 
*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x 
^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^(1/3))], -7 + 4*S 
qrt[3]])/(3^(1/4)*Sqrt[-1 + x^3/(1 + c^2*x^2)^(3/2)]*Sqrt[-((1 - (1 - (c^2 
*x^2)/(1 + c^2*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (c^2*x^2)/(1 + c^2*x^2))^( 
1/3))^2)])))/(4*c^2*x)))/8))/(8*c)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 214

Int[((a_) + (b_.)*(x_)^2)^(-7/6), x_Symbol] :> Simp[1/((a + b*x^2)^(2/3)*(a 
/(a + b*x^2))^(2/3))   Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x 
^2]], x] /; FreeQ[{a, b}, x]
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/6), x_Symbol] :> Simp[3*(x/(2*(a + b*x^2)^(1/ 
6))), x] - Simp[a/2   Int[1/(a + b*x^2)^(7/6), x], x] /; FreeQ[{a, b}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

rule 6213

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

Maple [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int x \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )d x\]

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x \,d x } \] Input:

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

Output:

integral((c^2*x^2 + 1)^(1/3)*(b*x*arcsinh(c*x) + a*x), x)
 

Sympy [N/A]

Not integrable

Time = 5.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt [3]{c^{2} x^{2} + 1}\, dx \] Input:

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

Output:

Integral(x*(a + b*asinh(c*x))*(c**2*x**2 + 1)**(1/3), x)
 

Maxima [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.65 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x \,d x } \] Input:

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

Output:

3/64*b*((8*(c^2*x^2 + 1)^(4/3)*log(c*x + sqrt(c^2*x^2 + 1)) - 3*(c^2*x^2 + 
 1)^(4/3))/c^2 - 64*integrate(1/8*(c^2*x^2 + 1)^(1/3)/(c^2*x + sqrt(c^2*x^ 
2 + 1)*c), x)) + 3/8*(c^2*x^2 + 1)^(4/3)*a/c^2
 

Giac [F(-2)]

Exception generated. \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [N/A]

Not integrable

Time = 2.94 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{1/3} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 5.35 \[ \int x \sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {8 \left (\sqrt {c^{2} x^{2}+1}+c x \right )^{\frac {2}{3}} \left (\int \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}+3 \left (\sqrt {c^{2} x^{2}+1}\, c x +c^{2} x^{2}+1\right )^{\frac {2}{3}} a \,c^{2} x^{2}+3 \left (\sqrt {c^{2} x^{2}+1}\, c x +c^{2} x^{2}+1\right )^{\frac {2}{3}} a}{8 \left (\sqrt {c^{2} x^{2}+1}+c x \right )^{\frac {2}{3}} c^{2}} \] Input:

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

Output:

(8*(sqrt(c**2*x**2 + 1) + c*x)**(2/3)*int((c**2*x**2 + 1)**(1/3)*asinh(c*x 
)*x,x)*b*c**2 + 3*(sqrt(c**2*x**2 + 1)*c*x + c**2*x**2 + 1)**(2/3)*a*c**2* 
x**2 + 3*(sqrt(c**2*x**2 + 1)*c*x + c**2*x**2 + 1)**(2/3)*a)/(8*(sqrt(c**2 
*x**2 + 1) + c*x)**(2/3)*c**2)