\(\int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [290]

Optimal result

Integrand size = 28, antiderivative size = 383 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {16 a b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {298 b^2 \left (1+c^2 x^2\right )}{225 c^6 \sqrt {d+c^2 d x^2}}-\frac {76 b^2 \left (1+c^2 x^2\right )^2}{675 c^6 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^3}{125 c^6 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d} \]

[Out]

298/225*b^2*(c^2*x^2+1)/c^6/(c^2*d*x^2+d)^(1/2)-76/675*b^2*(c^2*x^2+1)^2/c^6/(c^2*d*x^2+d)^(1/2)+2/125*b^2*(c^
2*x^2+1)^3/c^6/(c^2*d*x^2+d)^(1/2)-16/15*a*b*x*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)-16/15*b^2*x*arcsinh(c
*x)*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)+8/45*b*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d
)^(1/2)-2/25*b*x^5*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+8/15*(a+b*arcsinh(c*x))^2*(c^2*d
*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4/d+1/5*x^4*(a+b*arcsinh(c*x))^2*(c^2*
d*x^2+d)^(1/2)/c^2/d

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5812, 5798, 5772, 267, 5776, 272, 45} \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {2 b x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{25 c \sqrt {c^2 d x^2+d}}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {8 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {8 b x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {c^2 d x^2+d}}-\frac {16 a b x \sqrt {c^2 x^2+1}}{15 c^5 \sqrt {c^2 d x^2+d}}-\frac {16 b^2 x \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{15 c^5 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )^3}{125 c^6 \sqrt {c^2 d x^2+d}}-\frac {76 b^2 \left (c^2 x^2+1\right )^2}{675 c^6 \sqrt {c^2 d x^2+d}}+\frac {298 b^2 \left (c^2 x^2+1\right )}{225 c^6 \sqrt {c^2 d x^2+d}} \]

[In]

Int[(x^5*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]

[Out]

(-16*a*b*x*Sqrt[1 + c^2*x^2])/(15*c^5*Sqrt[d + c^2*d*x^2]) + (298*b^2*(1 + c^2*x^2))/(225*c^6*Sqrt[d + c^2*d*x
^2]) - (76*b^2*(1 + c^2*x^2)^2)/(675*c^6*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2)^3)/(125*c^6*Sqrt[d + c^2*
d*x^2]) - (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(15*c^5*Sqrt[d + c^2*d*x^2]) + (8*b*x^3*Sqrt[1 + c^2*x^2]*
(a + b*ArcSinh[c*x]))/(45*c^3*Sqrt[d + c^2*d*x^2]) - (2*b*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c*Sq
rt[d + c^2*d*x^2]) + (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d + c^2*d*x^2]*(a
+ b*ArcSinh[c*x])^2)/(15*c^4*d) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(5*c^2*d)

Rule 45

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] && NeQ[b*c - a*d, 0] && 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 267

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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 5798

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] - Dist[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 5812

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] + (-Dist[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] - Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{5 c^2}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int x^4 (a+b \text {arcsinh}(c x)) \, dx}{5 c \sqrt {d+c^2 d x^2}} \\ & = -\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {8 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{15 c^4}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^5}{\sqrt {1+c^2 x^2}} \, dx}{25 \sqrt {d+c^2 d x^2}}+\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int x^2 (a+b \text {arcsinh}(c x)) \, dx}{15 c^3 \sqrt {d+c^2 d x^2}} \\ & = \frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b \sqrt {1+c^2 x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{15 c^5 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{45 c^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1+c^2 x}}-\frac {2 \sqrt {1+c^2 x}}{c^4}+\frac {\left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \text {arcsinh}(c x) \, dx}{15 c^5 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{45 c^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{25 c^6 \sqrt {d+c^2 d x^2}}-\frac {4 b^2 \left (1+c^2 x^2\right )^2}{75 c^6 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^3}{125 c^6 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{15 c^4 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{45 c^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {298 b^2 \left (1+c^2 x^2\right )}{225 c^6 \sqrt {d+c^2 d x^2}}-\frac {76 b^2 \left (1+c^2 x^2\right )^2}{675 c^6 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^3}{125 c^6 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.60 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {-30 a b c x \sqrt {1+c^2 x^2} \left (120-20 c^2 x^2+9 c^4 x^4\right )+225 a^2 \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right )+2 b^2 \left (2072+1936 c^2 x^2-109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {1+c^2 x^2} \left (-120+20 c^2 x^2-9 c^4 x^4\right )+15 a \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right )\right ) \text {arcsinh}(c x)+225 b^2 \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right ) \text {arcsinh}(c x)^2}{3375 c^6 \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]

[Out]

(-30*a*b*c*x*Sqrt[1 + c^2*x^2]*(120 - 20*c^2*x^2 + 9*c^4*x^4) + 225*a^2*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6)
+ 2*b^2*(2072 + 1936*c^2*x^2 - 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[1 + c^2*x^2]*(-120 + 20*c^2*x^2 -
9*c^4*x^4) + 15*a*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6))*ArcSinh[c*x] + 225*b^2*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c
^6*x^6)*ArcSinh[c*x]^2)/(3375*c^6*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1226\) vs. \(2(335)=670\).

Time = 0.33 (sec) , antiderivative size = 1227, normalized size of antiderivative = 3.20

method	result	size
default	\(\text {Expression too large to display}\)	\(1227\)
parts	\(\text {Expression too large to display}\)	\(1227\)

[In]

int(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(1/5*x^4/c^2/d*(c^2*d*x^2+d)^(1/2)-4/5/c^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2
)))+b^2*(1/4000*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+
1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2-10*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)-5/864*(
d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*
x)^2-6*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)+5/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh
(c*x)^2-2*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)+5/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcs
inh(c*x)^2+2*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)-5/864*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1
/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2+6*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)+1/4000*(d*(c^2*
x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*
x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2+10*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1))+2*a*b*(1/800*(d*(c^2*x^2+1))^(
1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^
2+1)^(1/2)+1)*(-1+5*arcsinh(c*x))/c^6/d/(c^2*x^2+1)-5/288*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+
1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))/c^6/d/(c^2*x^2+1)+5/16*(d*(c^2*x^2+1))^(1/2)
*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))/c^6/d/(c^2*x^2+1)+5/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x
*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)+1)/c^6/d/(c^2*x^2+1)-5/288*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2
*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(3*arcsinh(c*x)+1)/c^6/d/(c^2*x^2+1)+1/800*(d*(c^2*x^2+1))^
(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x
^2+1)^(1/2)+1)*(1+5*arcsinh(c*x))/c^6/d/(c^2*x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} - b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{6} x^{6} - 15 \, a b c^{4} x^{4} + 60 \, a b c^{2} x^{2} + 120 \, a b - {\left (9 \, b^{2} c^{5} x^{5} - 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} - {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} + 1800 \, a^{2} + 4144 \, b^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} - 20 \, a b c^{3} x^{3} + 120 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} + c^{6} d\right )}} \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/3375*(225*(3*b^2*c^6*x^6 - b^2*c^4*x^4 + 4*b^2*c^2*x^2 + 8*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 - 15*a*b*c^4*x^4 + 60*a*b*c^2*x^2 + 120*a*b - (9*b^2*c^5*x^5 - 20*b^2*c^3*x^3 + 12
0*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
- (225*a^2 + 218*b^2)*c^4*x^4 + 4*(225*a^2 + 968*b^2)*c^2*x^2 + 1800*a^2 + 4144*b^2 - 30*(9*a*b*c^5*x^5 - 20*a
*b*c^3*x^3 + 120*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^8*d*x^2 + c^6*d)

Sympy [F]

\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(x**5*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**5*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.92 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} + 1} c^{2} x^{4} - 136 \, \sqrt {c^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} + 1}}{c^{2}}}{c^{4} \sqrt {d}} - \frac {15 \, {\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} \operatorname {arsinh}\left (c x\right )}{c^{5} \sqrt {d}}\right )} - \frac {2 \, {\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} a b}{225 \, c^{5} \sqrt {d}} \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*b
^2*arcsinh(c*x)^2 + 2/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d
*x^2 + d)/(c^6*d))*a*b*arcsinh(c*x) + 1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4
*d) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*a^2 + 2/3375*b^2*((27*sqrt(c^2*x^2 + 1)*c^2*x^4 - 136*sqrt(c^2*x^2 + 1)*x
^2 + 2072*sqrt(c^2*x^2 + 1)/c^2)/(c^4*sqrt(d)) - 15*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*arcsinh(c*x)/(c^5*sqrt(d)
)) - 2/225*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*a*b/(c^5*sqrt(d))

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \]

[In]

int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2),x)

[Out]

int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2), x)