3.290 \(\int \frac{x^5 (a+b \sinh ^{-1}(c x))^2}{\sqrt{d+c^2 d x^2}} \, dx\)

Optimal. Leaf size=383 \[ -\frac{16 a b x \sqrt{c^2 x^2+1}}{15 c^5 \sqrt{c^2 d x^2+d}}-\frac{2 b x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{c^2 d x^2+d}}+\frac{x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}+\frac{2 b^2 \left (c^2 x^2+1\right )^3}{125 c^6 \sqrt{c^2 d x^2+d}}-\frac{76 b^2 \left (c^2 x^2+1\right )^2}{675 c^6 \sqrt{c^2 d x^2+d}}+\frac{298 b^2 \left (c^2 x^2+1\right )}{225 c^6 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{15 c^5 \sqrt{c^2 d x^2+d}} \]

[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)

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Rubi [A]  time = 0.554273, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5758, 5717, 5653, 261, 5661, 266, 43} \[ -\frac{16 a b x \sqrt{c^2 x^2+1}}{15 c^5 \sqrt{c^2 d x^2+d}}-\frac{2 b x^5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{c^2 d x^2+d}}+\frac{x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{8 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}+\frac{2 b^2 \left (c^2 x^2+1\right )^3}{125 c^6 \sqrt{c^2 d x^2+d}}-\frac{76 b^2 \left (c^2 x^2+1\right )^2}{675 c^6 \sqrt{c^2 d x^2+d}}+\frac{298 b^2 \left (c^2 x^2+1\right )}{225 c^6 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{15 c^5 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[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 5758

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

Rule 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 5653

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 261

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 5661

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 266

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 43

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])

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx &=\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{4 \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^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 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{5 c \sqrt{d+c^2 d x^2}}\\ &=-\frac{2 b x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{d+c^2 d x^2}}-\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^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 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt{d+c^2 d x^2}}\\ &=\frac{8 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{d+c^2 d x^2}}+\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{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 \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{d+c^2 d x^2}}+\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{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 \sinh ^{-1}(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 ) \operatorname{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} \sinh ^{-1}(c x)}{15 c^5 \sqrt{d+c^2 d x^2}}+\frac{8 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{d+c^2 d x^2}}+\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \operatorname{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} \sinh ^{-1}(c x)}{15 c^5 \sqrt{d+c^2 d x^2}}+\frac{8 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c \sqrt{d+c^2 d x^2}}+\frac{8 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{15 c^4 d}+\frac{x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.365514, size = 230, normalized size = 0.6 \[ \frac{225 a^2 \left (3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right )-30 a b c x \sqrt{c^2 x^2+1} \left (9 c^4 x^4-20 c^2 x^2+120\right )+30 b \sinh ^{-1}(c x) \left (15 a \left (3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right )+b c x \sqrt{c^2 x^2+1} \left (-9 c^4 x^4+20 c^2 x^2-120\right )\right )+2 b^2 \left (27 c^6 x^6-109 c^4 x^4+1936 c^2 x^2+2072\right )+225 b^2 \left (3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right ) \sinh ^{-1}(c x)^2}{3375 c^6 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.355, size = 1227, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*(1+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)+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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.31596, size = 711, normalized size = 1.86 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{\sqrt{c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^5/sqrt(c^2*d*x^2 + d), x)