\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [309]

Optimal result

Integrand size = 28, antiderivative size = 293 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {2 b^2 \sqrt {d+c^2 d x^2}}{c^4 d^2}-\frac {2 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{c^4 d^2}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^4 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^4 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^4 d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a^2+2 b^2+a^2 c^2 x^2+2 b^2 c^2 x^2-2 a b c x \sqrt {1+c^2 x^2}+4 a b \text {arcsinh}(c x)+2 a b c^2 x^2 \text {arcsinh}(c x)-2 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+2 b^2 \text {arcsinh}(c x)^2+b^2 c^2 x^2 \text {arcsinh}(c x)^2-4 a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )}{c^4 d \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6225, 6213, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((x^2*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) + (2*((Sqrt[d 
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*b*Sqrt[1 + c^2*x^2]*(a*x 
 - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/(c*Sqrt[d + c^2*d*x^2])))/ 
(c^2*d) + (2*b*Sqrt[1 + c^2*x^2]*(-((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*d*Sqrt[d + c^2*d*x^2])
 

Defintions of rubi rules used

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 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 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]
 

rule 6225

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(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]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 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]
 

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.41

method	result	size
default	\(a^{2} \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 \operatorname {arcsinh}\left (x c \right )^{2}+2\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}\)	\(412\)
parts	\(a^{2} \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 \operatorname {arcsinh}\left (x c \right )^{2}+2\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}\)	\(412\)

Input:

int(x^3*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a^2*(x^2/c^2/d/(c^2*d*x^2+d)^(1/2)+2/d/c^4/(c^2*d*x^2+d)^(1/2))+b^2*(d*(c^ 
2*x^2+1))^(1/2)*(arcsinh(x*c)^2*x^2*c^2-2*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c) 
*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))+2*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+I 
*(x*c+(c^2*x^2+1)^(1/2)))-2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+2*c^2*x^2+2 
*I*(c^2*x^2+1)^(1/2)*dilog(1+I*(x*c+(c^2*x^2+1)^(1/2)))-2*I*(c^2*x^2+1)^(1 
/2)*dilog(1-I*(x*c+(c^2*x^2+1)^(1/2)))+2*arcsinh(x*c)^2+2)/d^2/c^4/(c^2*x^ 
2+1)+2*a*b*(d*(c^2*x^2+1))^(1/2)*(arcsinh(x*c)*c^2*x^2+I*(c^2*x^2+1)^(1/2) 
*ln(x*c+(c^2*x^2+1)^(1/2)-I)-I*(c^2*x^2+1)^(1/2)*ln(x*c+(c^2*x^2+1)^(1/2)+ 
I)-(c^2*x^2+1)^(1/2)*x*c+2*arcsinh(x*c))/d^2/c^4/(c^2*x^2+1)
 

Fricas [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

integral((b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3)*sqrt( 
c^2*d*x^2 + d)/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

-2*a*b*c*(x/(c^4*d^(3/2)) + arctan(c*x)/(c^5*d^(3/2))) + 2*a*b*(x^2/(sqrt( 
c^2*d*x^2 + d)*c^2*d) + 2/(sqrt(c^2*d*x^2 + d)*c^4*d))*arcsinh(c*x) + a^2* 
(x^2/(sqrt(c^2*d*x^2 + d)*c^2*d) + 2/(sqrt(c^2*d*x^2 + d)*c^4*d)) + b^2*(( 
c^2*x^2 + 2)*log(c*x + sqrt(c^2*x^2 + 1))^2/(sqrt(c^2*x^2 + 1)*c^4*d^(3/2) 
) - integrate(2*(c^4*x^4 + 3*c^2*x^2 + (c^3*x^3 + 2*c*x)*sqrt(c^2*x^2 + 1) 
 + 2)*log(c*x + sqrt(c^2*x^2 + 1))/((c^5*d^(3/2)*x^2 + c^3*d^(3/2))*(c^2*x 
^2 + 1) + (c^6*d^(3/2)*x^3 + c^4*d^(3/2)*x)*sqrt(c^2*x^2 + 1)), x))
 

Giac [F(-2)]

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

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),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 [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, a^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{6} x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{4}+\left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{6} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{4}}{\sqrt {d}\, c^{4} d \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

(sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 + 2*sqrt(c**2*x**2 + 1)*a**2 + 2*int(( 
asinh(c*x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)* 
a*b*c**6*x**2 + 2*int((asinh(c*x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + s 
qrt(c**2*x**2 + 1)),x)*a*b*c**4 + int((asinh(c*x)**2*x**3)/(sqrt(c**2*x**2 
 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**6*x**2 + int((asinh(c*x) 
**2*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c* 
*4)/(sqrt(d)*c**4*d*(c**2*x**2 + 1))