\(\int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx\) [264]

Optimal result

Integrand size = 28, antiderivative size = 309 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {52 b^2 \sqrt {d+c^2 d x^2}}{225 c^4}-\frac {26 b^2 \left (d+c^2 d x^2\right )^{3/2}}{675 c^4 d}+\frac {2 b^2 \left (d+c^2 d x^2\right )^{5/2}}{125 c^4 d^2}+\frac {4 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \] Output:

-52/225*b^2*(c^2*d*x^2+d)^(1/2)/c^4-26/675*b^2*(c^2*d*x^2+d)^(3/2)/c^4/d+2 
/125*b^2*(c^2*d*x^2+d)^(5/2)/c^4/d^2+4/15*b*x*(c^2*d*x^2+d)^(1/2)*(a+b*arc 
sinh(c*x))/c^3/(c^2*x^2+1)^(1/2)-2/45*b*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsi 
nh(c*x))/c/(c^2*x^2+1)^(1/2)-2/25*b*c*x^5*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh 
(c*x))/(c^2*x^2+1)^(1/2)-2/15*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^4 
+1/15*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^2+1/5*x^4*(c^2*d*x^2+ 
d)^(1/2)*(a+b*arcsinh(c*x))^2
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.72 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d+c^2 d x^2} \left (225 \left (-2+3 c^2 x^2\right ) \left (a+a c^2 x^2\right )^2-30 a b c x \sqrt {1+c^2 x^2} \left (-30+5 c^2 x^2+9 c^4 x^4\right )+2 b^2 \left (-428-439 c^2 x^2+16 c^4 x^4+27 c^6 x^6\right )-30 b \left (-15 a \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (-30+5 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 \left (-2+3 c^2 x^2\right ) \left (b+b c^2 x^2\right )^2 \text {arcsinh}(c x)^2\right )}{3375 c^4 \left (1+c^2 x^2\right )} \] Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*(225*(-2 + 3*c^2*x^2)*(a + a*c^2*x^2)^2 - 30*a*b*c*x* 
Sqrt[1 + c^2*x^2]*(-30 + 5*c^2*x^2 + 9*c^4*x^4) + 2*b^2*(-428 - 439*c^2*x^ 
2 + 16*c^4*x^4 + 27*c^6*x^6) - 30*b*(-15*a*(1 + c^2*x^2)^2*(-2 + 3*c^2*x^2 
) + b*c*x*Sqrt[1 + c^2*x^2]*(-30 + 5*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 
225*(-2 + 3*c^2*x^2)*(b + b*c^2*x^2)^2*ArcSinh[c*x]^2))/(3375*c^4*(1 + c^2 
*x^2))
 

Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6221, 6191, 243, 53, 2009, 6227, 6191, 243, 53, 2009, 6213, 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.

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \left (\frac {\left (c^2 x^2+1\right )^{3/2}}{c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {1}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arcsinh}(c x))dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*Sqrt[d + c^2*d 
*x^2]*(-1/10*(b*c*((2*Sqrt[1 + c^2*x^2])/c^6 - (4*(1 + c^2*x^2)^(3/2))/(3* 
c^6) + (2*(1 + c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSinh[c*x]))/5))/ 
(5*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2]*((x^2*Sqrt[1 + c^2*x^2]*(a + 
b*ArcSinh[c*x])^2)/(3*c^2) - (2*b*(-1/6*(b*c*((-2*Sqrt[1 + c^2*x^2])/c^4 + 
 (2*(1 + c^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSinh[c*x]))/3))/(3*c) 
- (2*((Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt 
[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/c))/(3*c^2)))/(5*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

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 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 6191

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

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 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(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] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[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]
 

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.57

method	result	size
orering	\(\frac {\left (1647 c^{8} x^{8}+2131 c^{6} x^{6}-8610 c^{4} x^{4}-13060 c^{2} x^{2}-5136\right ) \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{3375 c^{6} \left (c^{2} x^{2}+1\right ) x^{2}}-\frac {4 \left (81 c^{6} x^{6}+40 c^{4} x^{4}-878 c^{2} x^{2}-642\right ) \left (3 x^{2} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {2 b c \,x^{3} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{3375 x^{4} c^{6}}+\frac {\left (27 c^{4} x^{4}-11 c^{2} x^{2}-428\right ) \left (c^{2} x^{2}+1\right ) \left (6 x \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {7 x^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {12 b c \,x^{2} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\sqrt {c^{2} x^{2}+1}}-\frac {x^{5} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {4 x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}+\frac {2 c^{2} b^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{c^{2} x^{2}+1}-\frac {2 b \,c^{3} x^{4} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3375 c^{6} x^{3}}\)	\(485\)
default	\(\text {Expression too large to display}\)	\(1162\)
parts	\(\text {Expression too large to display}\)	\(1162\)

Input:

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

Output:

1/3375*(1647*c^8*x^8+2131*c^6*x^6-8610*c^4*x^4-13060*c^2*x^2-5136)/c^6/(c^ 
2*x^2+1)/x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2-4/3375*(81*c^6*x^6+4 
0*c^4*x^4-878*c^2*x^2-642)/x^4/c^6*(3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh 
(x*c))^2+x^4/(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2*c^2*d+2*b*c*x^3*(c^2 
*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))/(c^2*x^2+1)^(1/2))+1/3375*(27*c^4*x^4-1 
1*c^2*x^2-428)/c^6*(c^2*x^2+1)/x^3*(6*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x 
*c))^2+7*x^3/(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2*c^2*d+12*b*c*x^2*(c^ 
2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))/(c^2*x^2+1)^(1/2)-x^5/(c^2*d*x^2+d)^(3 
/2)*(a+b*arcsinh(x*c))^2*c^4*d^2+4*x^4/(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x* 
c))*c^3*d*b/(c^2*x^2+1)^(1/2)+2*c^2*b^2*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1 
)-2*b*c^3*x^4*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))/(c^2*x^2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.02 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{6} x^{6} + 60 \, a b c^{4} x^{4} - 15 \, a b c^{2} x^{2} - 30 \, a b - {\left (9 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} + 4 \, {\left (225 \, a^{2} + 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} + 878 \, b^{2}\right )} c^{2} x^{2} - 450 \, a^{2} - 856 \, b^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} + 5 \, a b c^{3} x^{3} - 30 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} + c^{4}\right )}} \] Input:

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

Output:

1/3375*(225*(3*b^2*c^6*x^6 + 4*b^2*c^4*x^4 - b^2*c^2*x^2 - 2*b^2)*sqrt(c^2 
*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^6*x^6 + 60*a*b*c 
^4*x^4 - 15*a*b*c^2*x^2 - 30*a*b - (9*b^2*c^5*x^5 + 5*b^2*c^3*x^3 - 30*b^2 
*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) 
+ (27*(25*a^2 + 2*b^2)*c^6*x^6 + 4*(225*a^2 + 8*b^2)*c^4*x^4 - (225*a^2 + 
878*b^2)*c^2*x^2 - 450*a^2 - 856*b^2 - 30*(9*a*b*c^5*x^5 + 5*a*b*c^3*x^3 - 
 30*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.98 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{15} \, a b {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} + 1} c^{2} \sqrt {d} x^{4} - 11 \, \sqrt {c^{2} x^{2} + 1} \sqrt {d} x^{2} - \frac {428 \, \sqrt {c^{2} x^{2} + 1} \sqrt {d}}{c^{2}}}{c^{2}} - \frac {15 \, {\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} a b}{225 \, c^{3}} \] Input:

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

Output:

1/15*b^2*(3*(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c 
^4*d))*arcsinh(c*x)^2 + 2/15*a*b*(3*(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2* 
(c^2*d*x^2 + d)^(3/2)/(c^4*d))*arcsinh(c*x) + 1/15*a^2*(3*(c^2*d*x^2 + d)^ 
(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 2/3375*b^2*((27*sqr 
t(c^2*x^2 + 1)*c^2*sqrt(d)*x^4 - 11*sqrt(c^2*x^2 + 1)*sqrt(d)*x^2 - 428*sq 
rt(c^2*x^2 + 1)*sqrt(d)/c^2)/c^2 - 15*(9*c^4*sqrt(d)*x^5 + 5*c^2*sqrt(d)*x 
^3 - 30*sqrt(d)*x)*arcsinh(c*x)/c^3) - 2/225*(9*c^4*sqrt(d)*x^5 + 5*c^2*sq 
rt(d)*x^3 - 30*sqrt(d)*x)*a*b/c^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, \left (3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+\sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a^{2}+30 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+15 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}\right )}{15 c^{4}} \] Input:

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

Output:

(sqrt(d)*(3*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 + sqrt(c**2*x**2 + 1)*a**2* 
c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a**2 + 30*int(sqrt(c**2*x**2 + 1)*asinh( 
c*x)*x**3,x)*a*b*c**4 + 15*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**3,x)*b 
**2*c**4))/(15*c**4)