\(\int x^3 (d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2 \, dx\) [211]

Optimal result

Integrand size = 24, antiderivative size = 198 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {b^2 d x^2}{24 c^2}+\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{12 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {d (a+b \text {arcsinh}(c x))^2}{24 c^4}+\frac {1}{12} d x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \] Output:

-1/24*b^2*d*x^2/c^2+1/72*b^2*d*x^4+1/108*b^2*c^2*d*x^6+1/12*b*d*x*(c^2*x^2 
+1)^(1/2)*(a+b*arcsinh(c*x))/c^3-1/18*b*d*x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsi 
nh(c*x))/c-1/18*b*c*d*x^5*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))-1/24*d*(a+b 
*arcsinh(c*x))^2/c^4+1/12*d*x^4*(a+b*arcsinh(c*x))^2+1/6*d*x^4*(c^2*x^2+1) 
*(a+b*arcsinh(c*x))^2
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.94 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (c x \left (18 a^2 c^3 x^3 \left (3+2 c^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-3+2 c^2 x^2+2 c^4 x^4\right )+b^2 c x \left (-9+3 c^2 x^2+2 c^4 x^4\right )\right )+6 b \left (b c x \sqrt {1+c^2 x^2} \left (3-2 c^2 x^2-2 c^4 x^4\right )+3 a \left (-1+6 c^4 x^4+4 c^6 x^6\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (-1+6 c^4 x^4+4 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{216 c^4} \] Input:

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

Output:

(d*(c*x*(18*a^2*c^3*x^3*(3 + 2*c^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-3 + 2* 
c^2*x^2 + 2*c^4*x^4) + b^2*c*x*(-9 + 3*c^2*x^2 + 2*c^4*x^4)) + 6*b*(b*c*x* 
Sqrt[1 + c^2*x^2]*(3 - 2*c^2*x^2 - 2*c^4*x^4) + 3*a*(-1 + 6*c^4*x^4 + 4*c^ 
6*x^6))*ArcSinh[c*x] + 9*b^2*(-1 + 6*c^4*x^4 + 4*c^6*x^6)*ArcSinh[c*x]^2)) 
/(216*c^4)
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.66, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6223, 6191, 6221, 15, 6227, 15, 6227, 15, 6198}

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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c d \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{6} d x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {1}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )\)

Input:

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

Output:

(d*x^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/6 - (b*c*d*(-1/36*(b*c*x^6) + 
 (x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/6 + (-1/16*(b*x^4)/c + (x^3* 
Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x* 
Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/( 
4*b*c^3)))/(4*c^2))/6))/3 + (d*((x^4*(a + b*ArcSinh[c*x])^2)/4 - (b*c*(-1/ 
16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*( 
-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + 
 b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*c^2)))/2))/3
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 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 6223

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[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 = 1.61 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.34

method	result	size
parts	\(a^{2} d \left (\frac {1}{6} x^{6} c^{2}+\frac {1}{4} x^{4}\right )+\frac {b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{6}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{12}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{12}+\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{24}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{72}-\frac {c^{2} x^{2}}{24}-\frac {1}{24}\right )}{c^{4}}+\frac {2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}-\frac {\sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{36}\right )}{c^{4}}\)	\(266\)
derivativedivides	\(\frac {a^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{4}\right )+b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{6}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{12}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{12}+\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{24}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{72}-\frac {c^{2} x^{2}}{24}-\frac {1}{24}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}+\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}\right )}{c^{4}}\)	\(269\)
default	\(\frac {a^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{4}\right )+b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{6}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{12}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{18}+\frac {\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{12}+\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{24}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{72}-\frac {c^{2} x^{2}}{24}-\frac {1}{24}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {\operatorname {arcsinh}\left (x c \right )}{24}+\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{36}+\frac {\sqrt {c^{2} x^{2}+1}\, x c}{24}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}\right )}{c^{4}}\)	\(269\)
orering	\(\frac {\left (182 c^{8} x^{8}+473 c^{6} x^{6}+42 c^{4} x^{4}-369 c^{2} x^{2}-180\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{432 c^{4} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (10 c^{6} x^{6}+21 c^{4} x^{4}-23 c^{2} x^{2}-24\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+2 x^{4} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{144 x^{2} c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\left (2 c^{4} x^{4}+3 c^{2} x^{2}-9\right ) \left (6 x \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+14 x^{3} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {12 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+\frac {8 x^{4} c^{3} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{432 x \,c^{4}}\)	\(402\)

Input:

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

Output:

a^2*d*(1/6*x^6*c^2+1/4*x^4)+b^2*d/c^4*(1/6*(c^2*x^2+1)^2*x^2*c^2*arcsinh(x 
*c)^2-1/12*arcsinh(x*c)^2*(c^2*x^2+1)^2-1/18*arcsinh(x*c)*x*c*(c^2*x^2+1)^ 
(5/2)+1/18*arcsinh(x*c)*x*c*(c^2*x^2+1)^(3/2)+1/12*arcsinh(x*c)*(c^2*x^2+1 
)^(1/2)*x*c+1/24*arcsinh(x*c)^2+1/108*(c^2*x^2+1)^3-1/72*(c^2*x^2+1)^2-1/2 
4*c^2*x^2-1/24)+2*a*b*d/c^4*(1/6*arcsinh(x*c)*x^6*c^6+1/4*arcsinh(x*c)*c^4 
*x^4-1/36*(c^2*x^2+1)^(1/2)*c^3*x^3+1/24*(c^2*x^2+1)^(1/2)*x*c-1/24*arcsin 
h(x*c)-1/36*(c^2*x^2+1)^(1/2)*x^5*c^5)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.21 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2 \, {\left (18 \, a^{2} + b^{2}\right )} c^{6} d x^{6} + 3 \, {\left (18 \, a^{2} + b^{2}\right )} c^{4} d x^{4} - 9 \, b^{2} c^{2} d x^{2} + 9 \, {\left (4 \, b^{2} c^{6} d x^{6} + 6 \, b^{2} c^{4} d x^{4} - b^{2} d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (12 \, a b c^{6} d x^{6} + 18 \, a b c^{4} d x^{4} - 3 \, a b d - {\left (2 \, b^{2} c^{5} d x^{5} + 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (2 \, a b c^{5} d x^{5} + 2 \, a b c^{3} d x^{3} - 3 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{216 \, c^{4}} \] Input:

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

Output:

1/216*(2*(18*a^2 + b^2)*c^6*d*x^6 + 3*(18*a^2 + b^2)*c^4*d*x^4 - 9*b^2*c^2 
*d*x^2 + 9*(4*b^2*c^6*d*x^6 + 6*b^2*c^4*d*x^4 - b^2*d)*log(c*x + sqrt(c^2* 
x^2 + 1))^2 + 6*(12*a*b*c^6*d*x^6 + 18*a*b*c^4*d*x^4 - 3*a*b*d - (2*b^2*c^ 
5*d*x^5 + 2*b^2*c^3*d*x^3 - 3*b^2*c*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt 
(c^2*x^2 + 1)) - 6*(2*a*b*c^5*d*x^5 + 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x)*sqrt( 
c^2*x^2 + 1))/c^4
 

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.68 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{6}}{6} + \frac {a^{2} d x^{4}}{4} + \frac {a b c^{2} d x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {a b c d x^{5} \sqrt {c^{2} x^{2} + 1}}{18} + \frac {a b d x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {a b d x^{3} \sqrt {c^{2} x^{2} + 1}}{18 c} + \frac {a b d x \sqrt {c^{2} x^{2} + 1}}{12 c^{3}} - \frac {a b d \operatorname {asinh}{\left (c x \right )}}{12 c^{4}} + \frac {b^{2} c^{2} d x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{2} d x^{6}}{108} - \frac {b^{2} c d x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{18} + \frac {b^{2} d x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} d x^{4}}{72} - \frac {b^{2} d x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{18 c} - \frac {b^{2} d x^{2}}{24 c^{2}} + \frac {b^{2} d x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{12 c^{3}} - \frac {b^{2} d \operatorname {asinh}^{2}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c**2*d*x**6/6 + a**2*d*x**4/4 + a*b*c**2*d*x**6*asinh(c*x) 
/3 - a*b*c*d*x**5*sqrt(c**2*x**2 + 1)/18 + a*b*d*x**4*asinh(c*x)/2 - a*b*d 
*x**3*sqrt(c**2*x**2 + 1)/(18*c) + a*b*d*x*sqrt(c**2*x**2 + 1)/(12*c**3) - 
 a*b*d*asinh(c*x)/(12*c**4) + b**2*c**2*d*x**6*asinh(c*x)**2/6 + b**2*c**2 
*d*x**6/108 - b**2*c*d*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/18 + b**2*d*x** 
4*asinh(c*x)**2/4 + b**2*d*x**4/72 - b**2*d*x**3*sqrt(c**2*x**2 + 1)*asinh 
(c*x)/(18*c) - b**2*d*x**2/(24*c**2) + b**2*d*x*sqrt(c**2*x**2 + 1)*asinh( 
c*x)/(12*c**3) - b**2*d*asinh(c*x)**2/(24*c**4), Ne(c, 0)), (a**2*d*x**4/4 
, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (174) = 348\).

Time = 0.05 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.23 \[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{6} \, b^{2} c^{2} d x^{6} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{6} \, a^{2} c^{2} d x^{6} + \frac {1}{4} \, b^{2} d x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} d x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} a b c^{2} d + \frac {1}{864} \, {\left ({\left (\frac {8 \, x^{6}}{c^{2}} - \frac {15 \, x^{4}}{c^{4}} + \frac {45 \, x^{2}}{c^{6}} - \frac {45 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{8}}\right )} c^{2} - 6 \, {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b d + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} d \] Input:

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

Output:

1/6*b^2*c^2*d*x^6*arcsinh(c*x)^2 + 1/6*a^2*c^2*d*x^6 + 1/4*b^2*d*x^4*arcsi 
nh(c*x)^2 + 1/4*a^2*d*x^4 + 1/144*(48*x^6*arcsinh(c*x) - (8*sqrt(c^2*x^2 + 
 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 
15*arcsinh(c*x)/c^7)*c)*a*b*c^2*d + 1/864*((8*x^6/c^2 - 15*x^4/c^4 + 45*x^ 
2/c^6 - 45*log(c*x + sqrt(c^2*x^2 + 1))^2/c^8)*c^2 - 6*(8*sqrt(c^2*x^2 + 1 
)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15 
*arcsinh(c*x)/c^7)*c*arcsinh(c*x))*b^2*c^2*d + 1/16*(8*x^4*arcsinh(c*x) - 
(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/ 
c^5)*c)*a*b*d + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x^2 + 1) 
)^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 
+ 3*arcsinh(c*x)/c^5)*c*arcsinh(c*x))*b^2*d
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (12 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+18 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}-2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+3 \sqrt {c^{2} x^{2}+1}\, a b c x +36 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+36 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +6 a^{2} c^{6} x^{6}+9 a^{2} c^{4} x^{4}\right )}{36 c^{4}} \] Input:

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

Output:

(d*(12*asinh(c*x)*a*b*c**6*x**6 + 18*asinh(c*x)*a*b*c**4*x**4 - 2*sqrt(c** 
2*x**2 + 1)*a*b*c**5*x**5 - 2*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 + 3*sqrt(c 
**2*x**2 + 1)*a*b*c*x + 36*int(asinh(c*x)**2*x**5,x)*b**2*c**6 + 36*int(as 
inh(c*x)**2*x**3,x)*b**2*c**4 - 3*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 6*a 
**2*c**6*x**6 + 9*a**2*c**4*x**4))/(36*c**4)