3.1.28 \(\int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 24, antiderivative size = 156 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {4 b \sqrt {1+c^2 x^2}}{3 c^5 d}-\frac {b \left (1+c^2 x^2\right )^{3/2}}{9 c^5 d}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d} \]

output

-1/9*b*(c^2*x^2+1)^(3/2)/c^5/d-x*(a+b*arcsinh(c*x))/c^4/d+1/3*x^3*(a+b*arc 
sinh(c*x))/c^2/d+2*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c^5/d- 
I*b*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^5/d+I*b*polylog(2,I*(c*x+(c^2* 
x^2+1)^(1/2)))/c^5/d+4/3*b*(c^2*x^2+1)^(1/2)/c^5/d
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {-9 a c x+3 a c^3 x^3+11 b \sqrt {1+c^2 x^2}-b c^2 x^2 \sqrt {1+c^2 x^2}-9 b c x \text {arcsinh}(c x)+3 b c^3 x^3 \text {arcsinh}(c x)+9 a \arctan (c x)+9 i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-9 i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-9 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+9 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{9 c^5 d} \]

input

Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]
 

output

(-9*a*c*x + 3*a*c^3*x^3 + 11*b*Sqrt[1 + c^2*x^2] - b*c^2*x^2*Sqrt[1 + c^2* 
x^2] - 9*b*c*x*ArcSinh[c*x] + 3*b*c^3*x^3*ArcSinh[c*x] + 9*a*ArcTan[c*x] + 
 (9*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (9*I)*b*ArcSinh[c*x]*Log 
[1 + I*E^ArcSinh[c*x]] - (9*I)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (9*I)*b 
*PolyLog[2, I*E^ArcSinh[c*x]])/(9*c^5*d)
 

3.1.28.3 Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6227, 27, 243, 53, 2009, 6227, 241, 6204, 3042, 4668, 2715, 2838}

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.

\(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))}{c^2 d x^2+d} \, dx\)

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2 d}-\frac {b \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx}{3 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c^2 d}-\frac {b \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2}{6 c d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}\)

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c d}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {-\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c d}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c d}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}}{c^2 d}+\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )}{6 c d}\)

input

Int[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]
 

output

-1/6*(b*((-2*Sqrt[1 + c^2*x^2])/c^4 + (2*(1 + c^2*x^2)^(3/2))/(3*c^4)))/(c 
*d) + (x^3*(a + b*ArcSinh[c*x]))/(3*c^2*d) - (-((b*Sqrt[1 + c^2*x^2])/c^3) 
 + (x*(a + b*ArcSinh[c*x]))/c^2 - (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh 
[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[ 
c*x]])/c^3)/(c^2*d)
 

3.1.28.3.1 Defintions of rubi rules used

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 241

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

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]
 

3.1.28.4 Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {c^{3} x^{3}}{3}-c x +\arctan \left (c x \right )\right )}{d}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\operatorname {arcsinh}\left (c x \right ) c x +\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}+\frac {11 \sqrt {c^{2} x^{2}+1}}{9}+\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c^{5}}\)	\(208\)
default	\(\frac {\frac {a \left (\frac {c^{3} x^{3}}{3}-c x +\arctan \left (c x \right )\right )}{d}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\operatorname {arcsinh}\left (c x \right ) c x +\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}+\frac {11 \sqrt {c^{2} x^{2}+1}}{9}+\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c^{5}}\)	\(208\)
parts	\(\frac {a \left (\frac {\frac {1}{3} x^{3} c^{2}-x}{c^{4}}+\frac {\arctan \left (c x \right )}{c^{5}}\right )}{d}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\operatorname {arcsinh}\left (c x \right ) c x +\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{9}+\frac {11 \sqrt {c^{2} x^{2}+1}}{9}+\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d \,c^{5}}\)	\(215\)

input

int(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d),x,method=_RETURNVERBOSE)
 

output

1/c^5*(a/d*(1/3*c^3*x^3-c*x+arctan(c*x))+b/d*(1/3*arcsinh(c*x)*c^3*x^3-arc 
sinh(c*x)*c*x+arcsinh(c*x)*arctan(c*x)-1/9*c^2*x^2*(c^2*x^2+1)^(1/2)+11/9* 
(c^2*x^2+1)^(1/2)+arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c 
*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^ 
(1/2))+I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))))
 

3.1.28.5 Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{c^{2} d x^{2} + d} \,d x } \]

input

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d),x, algorithm="fricas")
 

output

integral((b*x^4*arcsinh(c*x) + a*x^4)/(c^2*d*x^2 + d), x)
 

3.1.28.6 Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {\int \frac {a x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]

input

integrate(x**4*(a+b*asinh(c*x))/(c**2*d*x**2+d),x)
 

output

(Integral(a*x**4/(c**2*x**2 + 1), x) + Integral(b*x**4*asinh(c*x)/(c**2*x* 
*2 + 1), x))/d
 

3.1.28.7 Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{c^{2} d x^{2} + d} \,d x } \]

input

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d),x, algorithm="maxima")
 

output

1/3*a*((c^2*x^3 - 3*x)/(c^4*d) + 3*arctan(c*x)/(c^5*d)) + b*integrate(x^4* 
log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^2 + d), x)
 

3.1.28.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d),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
 

3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \]

input

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2),x)
 

output

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2), x)