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

Optimal. Leaf size=199 \[ \frac{d^3 \left (c^2 x^2+1\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (c^2 x^2+1\right )^{9/2}}{100 c^3}+\frac{7 b d^3 x \left (c^2 x^2+1\right )^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left (c^2 x^2+1\right )^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left (c^2 x^2+1\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{c^2 x^2+1}}{5120 c^3}+\frac{49 b d^3 \sinh ^{-1}(c x)}{5120 c^4} \]

[Out]

(49*b*d^3*x*Sqrt[1 + c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 + c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 + c^2
*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 + c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 + c^2*x^2)^(9/2))/(100*c^3)
+ (49*b*d^3*ArcSinh[c*x])/(5120*c^4) - (d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/(8*c^4) + (d^3*(1 + c^2*x^2
)^5*(a + b*ArcSinh[c*x]))/(10*c^4)

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Rubi [A]  time = 0.168706, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {266, 43, 5730, 12, 388, 195, 215} \[ \frac{d^3 \left (c^2 x^2+1\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (c^2 x^2+1\right )^{9/2}}{100 c^3}+\frac{7 b d^3 x \left (c^2 x^2+1\right )^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left (c^2 x^2+1\right )^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left (c^2 x^2+1\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{c^2 x^2+1}}{5120 c^3}+\frac{49 b d^3 \sinh ^{-1}(c x)}{5120 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*b*d^3*x*Sqrt[1 + c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 + c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 + c^2
*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 + c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 + c^2*x^2)^(9/2))/(100*c^3)
+ (49*b*d^3*ArcSinh[c*x])/(5120*c^4) - (d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/(8*c^4) + (d^3*(1 + c^2*x^2
)^5*(a + b*ArcSinh[c*x]))/(10*c^4)

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])

Rule 5730

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^3 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac{d^3 \left (1+c^2 x^2\right )^{7/2} \left (-1+4 c^2 x^2\right )}{40 c^4} \, dx\\ &=-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \left (-1+4 c^2 x^2\right ) \, dx}{40 c^3}\\ &=-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (7 b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \, dx}{200 c^3}\\ &=\frac{7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{1600 c^3}\\ &=\frac{49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{1920 c^3}\\ &=\frac{49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \sqrt{1+c^2 x^2} \, dx}{2560 c^3}\\ &=\frac{49 b d^3 x \sqrt{1+c^2 x^2}}{5120 c^3}+\frac{49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{5120 c^3}\\ &=\frac{49 b d^3 x \sqrt{1+c^2 x^2}}{5120 c^3}+\frac{49 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1+c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1+c^2 x^2\right )^{9/2}}{100 c^3}+\frac{49 b d^3 \sinh ^{-1}(c x)}{5120 c^4}-\frac{d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1+c^2 x^2\right )^5 \left (a+b \sinh ^{-1}(c x)\right )}{10 c^4}\\ \end{align*}

Mathematica [A]  time = 0.116578, size = 139, normalized size = 0.7 \[ \frac{d^3 \left (1920 a c^4 x^4 \left (4 c^6 x^6+15 c^4 x^4+20 c^2 x^2+10\right )-b c x \sqrt{c^2 x^2+1} \left (768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right )+15 b \left (512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right ) \sinh ^{-1}(c x)\right )}{76800 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(1920*a*c^4*x^4*(10 + 20*c^2*x^2 + 15*c^4*x^4 + 4*c^6*x^6) - b*c*x*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2
+ 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8) + 15*b*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c
^10*x^10)*ArcSinh[c*x]))/(76800*c^4)

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Maple [A]  time = 0.013, size = 195, normalized size = 1. \begin{align*}{\frac{1}{{c}^{4}} \left ({d}^{3}a \left ({\frac{{c}^{10}{x}^{10}}{10}}+{\frac{3\,{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{6}{x}^{6}}{2}}+{\frac{{c}^{4}{x}^{4}}{4}} \right ) +{d}^{3}b \left ({\frac{{\it Arcsinh} \left ( cx \right ){c}^{10}{x}^{10}}{10}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{8}{x}^{8}}{8}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{6}}{2}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}}{4}}-{\frac{{c}^{9}{x}^{9}}{100}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{57\,{c}^{7}{x}^{7}}{1600}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{401\,{c}^{5}{x}^{5}}{9600}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{79\,{c}^{3}{x}^{3}}{7680}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{79\,cx}{5120}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{79\,{\it Arcsinh} \left ( cx \right ) }{5120}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(d^3*a*(1/10*c^10*x^10+3/8*c^8*x^8+1/2*c^6*x^6+1/4*c^4*x^4)+d^3*b*(1/10*arcsinh(c*x)*c^10*x^10+3/8*arcsi
nh(c*x)*c^8*x^8+1/2*arcsinh(c*x)*c^6*x^6+1/4*arcsinh(c*x)*c^4*x^4-1/100*c^9*x^9*(c^2*x^2+1)^(1/2)-57/1600*c^7*
x^7*(c^2*x^2+1)^(1/2)-401/9600*c^5*x^5*(c^2*x^2+1)^(1/2)-79/7680*c^3*x^3*(c^2*x^2+1)^(1/2)+79/5120*c*x*(c^2*x^
2+1)^(1/2)-79/5120*arcsinh(c*x)))

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Maxima [B]  time = 1.1828, size = 644, normalized size = 3.24 \begin{align*} \frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} + \frac{1}{2} \, a c^{2} d^{3} x^{6} + \frac{1}{12800} \,{\left (1280 \, x^{10} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{128 \, \sqrt{c^{2} x^{2} + 1} x^{9}}{c^{2}} - \frac{144 \, \sqrt{c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac{168 \, \sqrt{c^{2} x^{2} + 1} x^{5}}{c^{6}} - \frac{210 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac{315 \, \sqrt{c^{2} x^{2} + 1} x}{c^{10}} - \frac{315 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\left (384 \, x^{8} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{48 \, \sqrt{c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac{56 \, \sqrt{c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac{105 \, \sqrt{c^{2} x^{2} + 1} x}{c^{8}} + \frac{105 \, \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{96} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{32} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 + 1/2*a*c^2*d^3*x^6 + 1/12800*(1280*x^10*arcsinh(c*x) - (128*sqrt(c^2*
x^2 + 1)*x^9/c^2 - 144*sqrt(c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(c^2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 + 1)*x^3/c
^8 + 315*sqrt(c^2*x^2 + 1)*x/c^10 - 315*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^10))*c)*b*c^6*d^3 + 1/1024*(384*
x^8*arcsinh(c*x) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6
- 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c)*b*c^4*d^3 + 1/4*a*d^3*x^4 +
1/96*(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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c)*b*c^2*d^3 + 1/32*(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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^3

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Fricas [A]  time = 2.51547, size = 474, normalized size = 2.38 \begin{align*} \frac{7680 \, a c^{10} d^{3} x^{10} + 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} + 19200 \, a c^{4} d^{3} x^{4} + 15 \,{\left (512 \, b c^{10} d^{3} x^{10} + 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} + 1280 \, b c^{4} d^{3} x^{4} - 79 \, b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (768 \, b c^{9} d^{3} x^{9} + 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} + 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}}{76800 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76800*(7680*a*c^10*d^3*x^10 + 28800*a*c^8*d^3*x^8 + 38400*a*c^6*d^3*x^6 + 19200*a*c^4*d^3*x^4 + 15*(512*b*c^
10*d^3*x^10 + 1920*b*c^8*d^3*x^8 + 2560*b*c^6*d^3*x^6 + 1280*b*c^4*d^3*x^4 - 79*b*d^3)*log(c*x + sqrt(c^2*x^2
+ 1)) - (768*b*c^9*d^3*x^9 + 2736*b*c^7*d^3*x^7 + 3208*b*c^5*d^3*x^5 + 790*b*c^3*d^3*x^3 - 1185*b*c*d^3*x)*sqr
t(c^2*x^2 + 1))/c^4

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Sympy [A]  time = 42.7176, size = 280, normalized size = 1.41 \begin{align*} \begin{cases} \frac{a c^{6} d^{3} x^{10}}{10} + \frac{3 a c^{4} d^{3} x^{8}}{8} + \frac{a c^{2} d^{3} x^{6}}{2} + \frac{a d^{3} x^{4}}{4} + \frac{b c^{6} d^{3} x^{10} \operatorname{asinh}{\left (c x \right )}}{10} - \frac{b c^{5} d^{3} x^{9} \sqrt{c^{2} x^{2} + 1}}{100} + \frac{3 b c^{4} d^{3} x^{8} \operatorname{asinh}{\left (c x \right )}}{8} - \frac{57 b c^{3} d^{3} x^{7} \sqrt{c^{2} x^{2} + 1}}{1600} + \frac{b c^{2} d^{3} x^{6} \operatorname{asinh}{\left (c x \right )}}{2} - \frac{401 b c d^{3} x^{5} \sqrt{c^{2} x^{2} + 1}}{9600} + \frac{b d^{3} x^{4} \operatorname{asinh}{\left (c x \right )}}{4} - \frac{79 b d^{3} x^{3} \sqrt{c^{2} x^{2} + 1}}{7680 c} + \frac{79 b d^{3} x \sqrt{c^{2} x^{2} + 1}}{5120 c^{3}} - \frac{79 b d^{3} \operatorname{asinh}{\left (c x \right )}}{5120 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**6*d**3*x**10/10 + 3*a*c**4*d**3*x**8/8 + a*c**2*d**3*x**6/2 + a*d**3*x**4/4 + b*c**6*d**3*x**1
0*asinh(c*x)/10 - b*c**5*d**3*x**9*sqrt(c**2*x**2 + 1)/100 + 3*b*c**4*d**3*x**8*asinh(c*x)/8 - 57*b*c**3*d**3*
x**7*sqrt(c**2*x**2 + 1)/1600 + b*c**2*d**3*x**6*asinh(c*x)/2 - 401*b*c*d**3*x**5*sqrt(c**2*x**2 + 1)/9600 + b
*d**3*x**4*asinh(c*x)/4 - 79*b*d**3*x**3*sqrt(c**2*x**2 + 1)/(7680*c) + 79*b*d**3*x*sqrt(c**2*x**2 + 1)/(5120*
c**3) - 79*b*d**3*asinh(c*x)/(5120*c**4), Ne(c, 0)), (a*d**3*x**4/4, True))

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Giac [B]  time = 1.89606, size = 633, normalized size = 3.18 \begin{align*} \frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} + \frac{1}{2} \, a c^{2} d^{3} x^{6} + \frac{1}{12800} \,{\left (1280 \, x^{10} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (\sqrt{c^{2} x^{2} + 1}{\left (2 \,{\left (4 \,{\left (2 \, x^{2}{\left (\frac{8 \, x^{2}}{c^{2}} - \frac{9}{c^{4}}\right )} + \frac{21}{c^{6}}\right )} x^{2} - \frac{105}{c^{8}}\right )} x^{2} + \frac{315}{c^{10}}\right )} x + \frac{315 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1} \right |}\right )}{c^{10}{\left | c \right |}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (\sqrt{c^{2} x^{2} + 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} - \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} - \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{96} \,{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (\sqrt{c^{2} x^{2} + 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} - \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x + \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (\sqrt{c^{2} x^{2} + 1} x{\left (\frac{2 \, x^{2}}{c^{2}} - \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 + 1/2*a*c^2*d^3*x^6 + 1/12800*(1280*x^10*log(c*x + sqrt(c^2*x^2 + 1))
- (sqrt(c^2*x^2 + 1)*(2*(4*(2*x^2*(8*x^2/c^2 - 9/c^4) + 21/c^6)*x^2 - 105/c^8)*x^2 + 315/c^10)*x + 315*log(abs
(-x*abs(c) + sqrt(c^2*x^2 + 1)))/(c^10*abs(c)))*c)*b*c^6*d^3 + 1/1024*(384*x^8*log(c*x + sqrt(c^2*x^2 + 1)) -
(sqrt(c^2*x^2 + 1)*(2*(4*x^2*(6*x^2/c^2 - 7/c^4) + 35/c^6)*x^2 - 105/c^8)*x - 105*log(abs(-x*abs(c) + sqrt(c^2
*x^2 + 1)))/(c^8*abs(c)))*c)*b*c^4*d^3 + 1/4*a*d^3*x^4 + 1/96*(48*x^6*log(c*x + sqrt(c^2*x^2 + 1)) - (sqrt(c^2
*x^2 + 1)*(2*x^2*(4*x^2/c^2 - 5/c^4) + 15/c^6)*x + 15*log(abs(-x*abs(c) + sqrt(c^2*x^2 + 1)))/(c^6*abs(c)))*c)
*b*c^2*d^3 + 1/32*(8*x^4*log(c*x + sqrt(c^2*x^2 + 1)) - (sqrt(c^2*x^2 + 1)*x*(2*x^2/c^2 - 3/c^4) - 3*log(abs(-
x*abs(c) + sqrt(c^2*x^2 + 1)))/(c^4*abs(c)))*c)*b*d^3