3.220 \(\int (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=291 \[ \frac{1}{7} d^3 x \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}-\frac{12 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac{32 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}+\frac{16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}+\frac{4322 b^2 d^3 x}{3675} \]

[Out]

(4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 + (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 -
(32*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(35*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))
/(105*c) - (12*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(175*c) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*A
rcSinh[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSinh[c*x])^2)/35 + (8*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3
5 + (6*d^3*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/35 + (d^3*x*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/7

________________________________________________________________________________________

Rubi [A]  time = 0.403062, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5684, 5653, 5717, 8, 194} \[ \frac{1}{7} d^3 x \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}-\frac{12 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac{32 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}+\frac{16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}+\frac{4322 b^2 d^3 x}{3675} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 + (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 -
(32*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(35*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))
/(105*c) - (12*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(175*c) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*A
rcSinh[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSinh[c*x])^2)/35 + (8*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3
5 + (6*d^3*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/35 + (d^3*x*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/7

Rule 5684

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*
(a + b*ArcSinh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^
n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1
+ c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Gt
Q[n, 0] && GtQ[p, 0]

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 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 8

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} (6 d) \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{7} \left (2 b c d^3\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{35} \left (24 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{1}{49} \left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \, dx-\frac{1}{35} \left (12 b c d^3\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{35} \left (16 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{1}{49} \left (2 b^2 d^3\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx+\frac{1}{175} \left (12 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac{1}{35} \left (16 b c d^3\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac{2}{49} b^2 d^3 x+\frac{2}{49} b^2 c^2 d^3 x^3+\frac{6}{245} b^2 c^4 d^3 x^5+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac{12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{175} \left (12 b^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{105} \left (16 b^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{35} \left (32 b c d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{962 b^2 d^3 x}{3675}+\frac{1514 b^2 c^2 d^3 x^3}{11025}+\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{32 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac{12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{35} \left (32 b^2 d^3\right ) \int 1 \, dx\\ &=\frac{4322 b^2 d^3 x}{3675}+\frac{1514 b^2 c^2 d^3 x^3}{11025}+\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{32 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}-\frac{16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac{12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.535683, size = 239, normalized size = 0.82 \[ \frac{d^3 \left (11025 a^2 c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )-210 a b \sqrt{c^2 x^2+1} \left (75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right )-210 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right )-105 a c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )\right )+2 b^2 c x \left (1125 c^6 x^6+7371 c^4 x^4+26495 c^2 x^2+226905\right )+11025 b^2 c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right ) \sinh ^{-1}(c x)^2\right )}{385875 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(11025*a^2*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - 210*a*b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2
+ 351*c^4*x^4 + 75*c^6*x^6) + 2*b^2*c*x*(226905 + 26495*c^2*x^2 + 7371*c^4*x^4 + 1125*c^6*x^6) - 210*b*(-105*
a*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*
c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]^2))/(385875*c)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 372, normalized size = 1.3 \begin{align*}{\frac{1}{c} \left ({d}^{3}{a}^{2} \left ({\frac{{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}+cx \right ) +{d}^{3}{b}^{2} \left ({\frac{16\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx}{35}}+{\frac{ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{7}}+{\frac{6\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{35}}+{\frac{8\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx \left ({c}^{2}{x}^{2}+1 \right ) }{35}}-{\frac{4322\,{\it Arcsinh} \left ( cx \right ) }{3675}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{413312\,cx}{385875}}-{\frac{2\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{49} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{134\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{1225} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{962\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}}{3675}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{3}}{343}}+{\frac{888\,cx \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}{42875}}+{\frac{30256\,cx \left ({c}^{2}{x}^{2}+1 \right ) }{385875}} \right ) +2\,{d}^{3}ab \left ( 1/7\,{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}+3/5\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}+{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}+{\it Arcsinh} \left ( cx \right ) cx-1/49\,{c}^{6}{x}^{6}\sqrt{{c}^{2}{x}^{2}+1}-{\frac{117\,{c}^{4}{x}^{4}\sqrt{{c}^{2}{x}^{2}+1}}{1225}}-{\frac{757\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}}{3675}}-{\frac{2161\,\sqrt{{c}^{2}{x}^{2}+1}}{3675}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(d^3*a^2*(1/7*c^7*x^7+3/5*c^5*x^5+c^3*x^3+c*x)+d^3*b^2*(16/35*arcsinh(c*x)^2*c*x+1/7*arcsinh(c*x)^2*c*x*(c
^2*x^2+1)^3+6/35*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2+8/35*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-4322/3675*arcsinh(c*x)*(
c^2*x^2+1)^(1/2)+413312/385875*c*x-2/49*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(5/2)-134/1225*arcsinh(c*x)*c^2*x^2*(
c^2*x^2+1)^(3/2)-962/3675*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+2/343*c*x*(c^2*x^2+1)^3+888/42875*c*x*(c^2*x^
2+1)^2+30256/385875*c*x*(c^2*x^2+1))+2*d^3*a*b*(1/7*arcsinh(c*x)*c^7*x^7+3/5*arcsinh(c*x)*c^5*x^5+arcsinh(c*x)
*c^3*x^3+arcsinh(c*x)*c*x-1/49*c^6*x^6*(c^2*x^2+1)^(1/2)-117/1225*c^4*x^4*(c^2*x^2+1)^(1/2)-757/3675*c^2*x^2*(
c^2*x^2+1)^(1/2)-2161/3675*(c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [B]  time = 1.29658, size = 961, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*c^6*d^3*x^7*arcsinh(c*x)^2 + 1/7*a^2*c^6*d^3*x^7 + 3/5*b^2*c^4*d^3*x^5*arcsinh(c*x)^2 + 3/5*a^2*c^4*d^
3*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^6*d^3 - 2/25725*(105*(5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqr
t(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^6*d^3 + b^2*c^2*d^3*x^3*arcsinh(c*x)^2 + 2/25*(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*c^4*d^3 -
2/375*(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*c^4*d^3 + a^2*c^2*d^3*x^3 + 2/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^
2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^2*d^3 - 2/9*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*
x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*c^2*d^3 + b^2*d^3*x*arcsinh(c*x)^2 + 2*b^2*d^3*(x - sqrt
(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^3*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^3/c

________________________________________________________________________________________

Fricas [A]  time = 2.78674, size = 818, normalized size = 2.81 \begin{align*} \frac{1125 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \,{\left (1225 \, a^{2} + 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \,{\left (11025 \, a^{2} + 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} + 105 \,{\left (3675 \, a^{2} + 4322 \, b^{2}\right )} c d^{3} x + 11025 \,{\left (5 \, b^{2} c^{7} d^{3} x^{7} + 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} + 35 \, b^{2} c d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 210 \,{\left (525 \, a b c^{7} d^{3} x^{7} + 2205 \, a b c^{5} d^{3} x^{5} + 3675 \, a b c^{3} d^{3} x^{3} + 3675 \, a b c d^{3} x -{\left (75 \, b^{2} c^{6} d^{3} x^{6} + 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} + 2161 \, b^{2} d^{3}\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^{3} x^{6} + 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} + 2161 \, a b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{385875 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/385875*(1125*(49*a^2 + 2*b^2)*c^7*d^3*x^7 + 189*(1225*a^2 + 78*b^2)*c^5*d^3*x^5 + 35*(11025*a^2 + 1514*b^2)*
c^3*d^3*x^3 + 105*(3675*a^2 + 4322*b^2)*c*d^3*x + 11025*(5*b^2*c^7*d^3*x^7 + 21*b^2*c^5*d^3*x^5 + 35*b^2*c^3*d
^3*x^3 + 35*b^2*c*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 210*(525*a*b*c^7*d^3*x^7 + 2205*a*b*c^5*d^3*x^5 + 36
75*a*b*c^3*d^3*x^3 + 3675*a*b*c*d^3*x - (75*b^2*c^6*d^3*x^6 + 351*b^2*c^4*d^3*x^4 + 757*b^2*c^2*d^3*x^2 + 2161
*b^2*d^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 210*(75*a*b*c^6*d^3*x^6 + 351*a*b*c^4*d^3*x^4 + 75
7*a*b*c^2*d^3*x^2 + 2161*a*b*d^3)*sqrt(c^2*x^2 + 1))/c

________________________________________________________________________________________

Sympy [A]  time = 18.6535, size = 524, normalized size = 1.8 \begin{align*} \begin{cases} \frac{a^{2} c^{6} d^{3} x^{7}}{7} + \frac{3 a^{2} c^{4} d^{3} x^{5}}{5} + a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x + \frac{2 a b c^{6} d^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{2 a b c^{5} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{49} + \frac{6 a b c^{4} d^{3} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{234 a b c^{3} d^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{1225} + 2 a b c^{2} d^{3} x^{3} \operatorname{asinh}{\left (c x \right )} - \frac{1514 a b c d^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname{asinh}{\left (c x \right )} - \frac{4322 a b d^{3} \sqrt{c^{2} x^{2} + 1}}{3675 c} + \frac{b^{2} c^{6} d^{3} x^{7} \operatorname{asinh}^{2}{\left (c x \right )}}{7} + \frac{2 b^{2} c^{6} d^{3} x^{7}}{343} - \frac{2 b^{2} c^{5} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{49} + \frac{3 b^{2} c^{4} d^{3} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}}{5} + \frac{234 b^{2} c^{4} d^{3} x^{5}}{6125} - \frac{234 b^{2} c^{3} d^{3} x^{4} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{1225} + b^{2} c^{2} d^{3} x^{3} \operatorname{asinh}^{2}{\left (c x \right )} + \frac{1514 b^{2} c^{2} d^{3} x^{3}}{11025} - \frac{1514 b^{2} c d^{3} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname{asinh}^{2}{\left (c x \right )} + \frac{4322 b^{2} d^{3} x}{3675} - \frac{4322 b^{2} d^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{3675 c} & \text{for}\: c \neq 0 \\a^{2} d^{3} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**6*d**3*x**7/7 + 3*a**2*c**4*d**3*x**5/5 + a**2*c**2*d**3*x**3 + a**2*d**3*x + 2*a*b*c**6*d*
*3*x**7*asinh(c*x)/7 - 2*a*b*c**5*d**3*x**6*sqrt(c**2*x**2 + 1)/49 + 6*a*b*c**4*d**3*x**5*asinh(c*x)/5 - 234*a
*b*c**3*d**3*x**4*sqrt(c**2*x**2 + 1)/1225 + 2*a*b*c**2*d**3*x**3*asinh(c*x) - 1514*a*b*c*d**3*x**2*sqrt(c**2*
x**2 + 1)/3675 + 2*a*b*d**3*x*asinh(c*x) - 4322*a*b*d**3*sqrt(c**2*x**2 + 1)/(3675*c) + b**2*c**6*d**3*x**7*as
inh(c*x)**2/7 + 2*b**2*c**6*d**3*x**7/343 - 2*b**2*c**5*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/49 + 3*b**2*c
**4*d**3*x**5*asinh(c*x)**2/5 + 234*b**2*c**4*d**3*x**5/6125 - 234*b**2*c**3*d**3*x**4*sqrt(c**2*x**2 + 1)*asi
nh(c*x)/1225 + b**2*c**2*d**3*x**3*asinh(c*x)**2 + 1514*b**2*c**2*d**3*x**3/11025 - 1514*b**2*c*d**3*x**2*sqrt
(c**2*x**2 + 1)*asinh(c*x)/3675 + b**2*d**3*x*asinh(c*x)**2 + 4322*b**2*d**3*x/3675 - 4322*b**2*d**3*sqrt(c**2
*x**2 + 1)*asinh(c*x)/(3675*c), Ne(c, 0)), (a**2*d**3*x, True))

________________________________________________________________________________________

Giac [B]  time = 2.98611, size = 1025, normalized size = 3.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*c^6*d^3*x^7 + 3/5*a^2*c^4*d^3*x^5 + 2/245*(35*x^7*log(c*x + sqrt(c^2*x^2 + 1)) - (5*(c^2*x^2 + 1)^(7/2
) - 21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2 + 1))/c^7)*a*b*c^6*d^3 + 1/25725*(3675*x
^7*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((75*c^6*x^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^7 - 105*(5*(c^2*x
^2 + 1)^(7/2) - 21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2
+ 1))/c^8))*b^2*c^6*d^3 + 2/25*(15*x^5*log(c*x + sqrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 +
1)^(3/2) + 15*sqrt(c^2*x^2 + 1))/c^5)*a*b*c^4*d^3 + 1/375*(225*x^5*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((9*c^
4*x^5 - 20*c^2*x^3 + 120*x)/c^5 - 15*(3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))*l
og(c*x + sqrt(c^2*x^2 + 1))/c^6))*b^2*c^4*d^3 + a^2*c^2*d^3*x^3 + 2/3*(3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) - ((
c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*a*b*c^2*d^3 + 1/9*(9*x^3*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*(
(c^2*x^3 - 6*x)/c^3 - 3*((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^4))*b^2*c^2
*d^3 + 2*(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*a*b*d^3 + (x*log(c*x + sqrt(c^2*x^2 + 1))^2 +
2*c*(x/c - sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1))/c^2))*b^2*d^3 + a^2*d^3*x