\(\int x^4 (d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2 \, dx\) [210]

Optimal result

Integrand size = 24, antiderivative size = 283 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {304 b^2 d x}{3675 c^4}-\frac {152 b^2 d x^3}{11025 c^2}+\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7-\frac {32 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{525 c^5}+\frac {16 b d x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{525 c^3}-\frac {4 b d x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{175 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{21 c^5}+\frac {4 b d \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{35 c^5}-\frac {2 b d \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{49 c^5}+\frac {2}{35} d x^5 (a+b \text {arcsinh}(c x))^2+\frac {1}{7} d x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2 \] Output:

304/3675*b^2*d*x/c^4-152/11025*b^2*d*x^3/c^2+38/6125*b^2*d*x^5+2/343*b^2*c 
^2*d*x^7-32/525*b*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^5+16/525*b*d*x^ 
2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3-4/175*b*d*x^4*(c^2*x^2+1)^(1/2) 
*(a+b*arcsinh(c*x))/c-2/21*b*d*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^5+4/ 
35*b*d*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^5-2/49*b*d*(c^2*x^2+1)^(7/2) 
*(a+b*arcsinh(c*x))/c^5+2/35*d*x^5*(a+b*arcsinh(c*x))^2+1/7*d*x^5*(c^2*x^2 
+1)*(a+b*arcsinh(c*x))^2
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.71 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (11025 a^2 c^5 x^5 \left (7+5 c^2 x^2\right )-210 a b \sqrt {1+c^2 x^2} \left (152-76 c^2 x^2+57 c^4 x^4+75 c^6 x^6\right )+b^2 \left (31920 c x-5320 c^3 x^3+2394 c^5 x^5+2250 c^7 x^7\right )-210 b \left (-105 a c^5 x^5 \left (7+5 c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (152-76 c^2 x^2+57 c^4 x^4+75 c^6 x^6\right )\right ) \text {arcsinh}(c x)+11025 b^2 c^5 x^5 \left (7+5 c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{385875 c^5} \] Input:

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

Output:

(d*(11025*a^2*c^5*x^5*(7 + 5*c^2*x^2) - 210*a*b*Sqrt[1 + c^2*x^2]*(152 - 7 
6*c^2*x^2 + 57*c^4*x^4 + 75*c^6*x^6) + b^2*(31920*c*x - 5320*c^3*x^3 + 239 
4*c^5*x^5 + 2250*c^7*x^7) - 210*b*(-105*a*c^5*x^5*(7 + 5*c^2*x^2) + b*Sqrt 
[1 + c^2*x^2]*(152 - 76*c^2*x^2 + 57*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 
 11025*b^2*c^5*x^5*(7 + 5*c^2*x^2)*ArcSinh[c*x]^2))/(385875*c^5)
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6223, 6191, 6219, 27, 2009, 6227, 15, 6227, 15, 6213, 24}

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.

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

\(\Big \downarrow \) 6223

\(\displaystyle -\frac {2}{7} b c d \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{7} d \int x^4 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c d \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6219

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c d \left (-b c \int \frac {15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8}{105 c^6}dx+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c d \left (-\frac {b \int \left (15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{105 c^5}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}-\frac {b \int x^4dx}{5 c}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}\right )\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{7} d x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {4 \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c d \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\)

Input:

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

Output:

(d*x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/7 - (2*b*c*d*(-1/105*(b*(8*x 
- (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (15*c^6*x^7)/7))/c^5 + ((1 + c^2*x^2)^(3 
/2)*(a + b*ArcSinh[c*x]))/(3*c^6) - (2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[ 
c*x]))/(5*c^6) + ((1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^6)))/7 + 
(2*d*((x^5*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*(-1/25*(b*x^5)/c + (x^4*Sqrt 
[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) - (4*(-1/9*(b*x^3)/c + (x^2*Sq 
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) - (2*(-((b*x)/c) + (Sqrt[1 + 
 c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2))/(3*c^2)))/(5*c^2)))/5))/7
 

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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

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 = 1.99 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.16

method	result	size
parts	\(a^{2} d \left (\frac {1}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{3} c^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{7}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{49}+\frac {62 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1225}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {37384 x c}{385875}-\frac {484 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {3358 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )}{c^{5}}+\frac {2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(329\)
derivativedivides	\(\frac {a^{2} d \left (\frac {1}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{3} c^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{7}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{49}+\frac {62 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1225}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {37384 x c}{385875}-\frac {484 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {3358 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(330\)
default	\(\frac {a^{2} d \left (\frac {1}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+b^{2} d \left (\frac {\left (c^{2} x^{2}+1\right )^{2} x^{3} c^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{7}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{49}+\frac {62 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1225}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {37384 x c}{385875}-\frac {484 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {3358 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 a b d \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {19 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(330\)
orering	\(\frac {\left (142875 c^{10} x^{10}+346302 c^{8} x^{8}+107235 c^{6} x^{6}+505400 c^{4} x^{4}+872480 c^{2} x^{2}+383040\right ) \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{385875 x \,c^{6} \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (20250 c^{8} x^{8}+36027 c^{6} x^{6}-21679 c^{4} x^{4}+162260 c^{2} x^{2}+143640\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+2 x^{5} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{385875 x^{4} c^{6} \left (c^{2} x^{2}+1\right )}+\frac {\left (1125 c^{6} x^{6}+1197 c^{4} x^{4}-2660 c^{2} x^{2}+15960\right ) \left (12 x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+18 x^{4} c^{2} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+\frac {16 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+\frac {8 x^{5} c^{3} d \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{5} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{385875 x^{3} c^{6}}\)	\(431\)

Input:

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

Output:

a^2*d*(1/7*c^2*x^7+1/5*x^5)+b^2*d/c^5*(1/7*(c^2*x^2+1)^2*x^3*c^3*arcsinh(x 
*c)^2-3/35*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^2+2/35*arcsinh(x*c)^2*x*c+1/35*a 
rcsinh(x*c)^2*x*c*(c^2*x^2+1)-2/49*arcsinh(x*c)*x^2*c^2*(c^2*x^2+1)^(5/2)+ 
62/1225*arcsinh(x*c)*(c^2*x^2+1)^(5/2)+2/343*x*c*(c^2*x^2+1)^3+37384/38587 
5*x*c-484/42875*x*c*(c^2*x^2+1)^2-3358/385875*x*c*(c^2*x^2+1)-4/35*arcsinh 
(x*c)*(c^2*x^2+1)^(1/2)-2/105*arcsinh(x*c)*(c^2*x^2+1)^(3/2))+2*a*b*d/c^5* 
(1/7*arcsinh(x*c)*c^7*x^7+1/5*arcsinh(x*c)*x^5*c^5-19/1225*x^4*c^4*(c^2*x^ 
2+1)^(1/2)+76/3675*x^2*c^2*(c^2*x^2+1)^(1/2)-152/3675*(c^2*x^2+1)^(1/2)-1/ 
49*x^6*c^6*(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.92 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d x^{7} + 63 \, {\left (1225 \, a^{2} + 38 \, b^{2}\right )} c^{5} d x^{5} - 5320 \, b^{2} c^{3} d x^{3} + 31920 \, b^{2} c d x + 11025 \, {\left (5 \, b^{2} c^{7} d x^{7} + 7 \, b^{2} c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{7} d x^{7} + 735 \, a b c^{5} d x^{5} - {\left (75 \, b^{2} c^{6} d x^{6} + 57 \, b^{2} c^{4} d x^{4} - 76 \, b^{2} c^{2} d x^{2} + 152 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 210 \, {\left (75 \, a b c^{6} d x^{6} + 57 \, a b c^{4} d x^{4} - 76 \, a b c^{2} d x^{2} + 152 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{385875 \, c^{5}} \] Input:

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

Output:

1/385875*(1125*(49*a^2 + 2*b^2)*c^7*d*x^7 + 63*(1225*a^2 + 38*b^2)*c^5*d*x 
^5 - 5320*b^2*c^3*d*x^3 + 31920*b^2*c*d*x + 11025*(5*b^2*c^7*d*x^7 + 7*b^2 
*c^5*d*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 210*(525*a*b*c^7*d*x^7 + 735* 
a*b*c^5*d*x^5 - (75*b^2*c^6*d*x^6 + 57*b^2*c^4*d*x^4 - 76*b^2*c^2*d*x^2 + 
152*b^2*d)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 210*(75*a*b*c 
^6*d*x^6 + 57*a*b*c^4*d*x^4 - 76*a*b*c^2*d*x^2 + 152*a*b*d)*sqrt(c^2*x^2 + 
 1))/c^5
 

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.37 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{7}}{7} + \frac {a^{2} d x^{5}}{5} + \frac {2 a b c^{2} d x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b c d x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {2 a b d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {38 a b d x^{4} \sqrt {c^{2} x^{2} + 1}}{1225 c} + \frac {152 a b d x^{2} \sqrt {c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {304 a b d \sqrt {c^{2} x^{2} + 1}}{3675 c^{5}} + \frac {b^{2} c^{2} d x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{2} d x^{7}}{343} - \frac {2 b^{2} c d x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49} + \frac {b^{2} d x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {38 b^{2} d x^{5}}{6125} - \frac {38 b^{2} d x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1225 c} - \frac {152 b^{2} d x^{3}}{11025 c^{2}} + \frac {152 b^{2} d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c^{3}} + \frac {304 b^{2} d x}{3675 c^{4}} - \frac {304 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{5}}{5} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c**2*d*x**7/7 + a**2*d*x**5/5 + 2*a*b*c**2*d*x**7*asinh(c* 
x)/7 - 2*a*b*c*d*x**6*sqrt(c**2*x**2 + 1)/49 + 2*a*b*d*x**5*asinh(c*x)/5 - 
 38*a*b*d*x**4*sqrt(c**2*x**2 + 1)/(1225*c) + 152*a*b*d*x**2*sqrt(c**2*x** 
2 + 1)/(3675*c**3) - 304*a*b*d*sqrt(c**2*x**2 + 1)/(3675*c**5) + b**2*c**2 
*d*x**7*asinh(c*x)**2/7 + 2*b**2*c**2*d*x**7/343 - 2*b**2*c*d*x**6*sqrt(c* 
*2*x**2 + 1)*asinh(c*x)/49 + b**2*d*x**5*asinh(c*x)**2/5 + 38*b**2*d*x**5/ 
6125 - 38*b**2*d*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(1225*c) - 152*b**2*d 
*x**3/(11025*c**2) + 152*b**2*d*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3675* 
c**3) + 304*b**2*d*x/(3675*c**4) - 304*b**2*d*sqrt(c**2*x**2 + 1)*asinh(c* 
x)/(3675*c**5), Ne(c, 0)), (a**2*d*x**5/5, True))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.56 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{7} \, b^{2} c^{2} d x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, b^{2} d x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} d x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d \] Input:

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

Output:

1/7*b^2*c^2*d*x^7*arcsinh(c*x)^2 + 1/7*a^2*c^2*d*x^7 + 1/5*b^2*d*x^5*arcsi 
nh(c*x)^2 + 1/5*a^2*d*x^5 + 2/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 
 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 
16*sqrt(c^2*x^2 + 1)/c^8)*c)*a*b*c^2*d - 2/25725*(105*(5*sqrt(c^2*x^2 + 1) 
*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16* 
sqrt(c^2*x^2 + 1)/c^8)*c*arcsinh(c*x) - (75*c^6*x^7 - 126*c^4*x^5 + 280*c^ 
2*x^3 - 1680*x)/c^6)*b^2*c^2*d + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x 
^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c 
)*a*b*d - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^ 
2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 
+ 120*x)/c^4)*b^2*d
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (1050 \mathit {asinh} \left (c x \right ) a b \,c^{7} x^{7}+1470 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}-150 \sqrt {c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-114 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}+152 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-304 \sqrt {c^{2} x^{2}+1}\, a b +3675 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+3675 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+525 a^{2} c^{7} x^{7}+735 a^{2} c^{5} x^{5}\right )}{3675 c^{5}} \] Input:

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

Output:

(d*(1050*asinh(c*x)*a*b*c**7*x**7 + 1470*asinh(c*x)*a*b*c**5*x**5 - 150*sq 
rt(c**2*x**2 + 1)*a*b*c**6*x**6 - 114*sqrt(c**2*x**2 + 1)*a*b*c**4*x**4 + 
152*sqrt(c**2*x**2 + 1)*a*b*c**2*x**2 - 304*sqrt(c**2*x**2 + 1)*a*b + 3675 
*int(asinh(c*x)**2*x**6,x)*b**2*c**7 + 3675*int(asinh(c*x)**2*x**4,x)*b**2 
*c**5 + 525*a**2*c**7*x**7 + 735*a**2*c**5*x**5))/(3675*c**5)