3.19 \(\int x^4 (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=226 \[ \frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{11/2}}{121 c^5}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{9/2}}{297 c^5}-\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{1617 c^5}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{1925 c^5}-\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{3465 c^5}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{1155 c^5} \]

[Out]

(-16*b*d^3*Sqrt[1 + c^2*x^2])/(1155*c^5) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(3465*c^5) - (2*b*d^3*(1 + c^2*x^2)^(
5/2))/(1925*c^5) - (b*d^3*(1 + c^2*x^2)^(7/2))/(1617*c^5) + (4*b*d^3*(1 + c^2*x^2)^(9/2))/(297*c^5) - (b*d^3*(
1 + c^2*x^2)^(11/2))/(121*c^5) + (d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^2*d^3*x^7*(a + b*ArcSinh[c*x]))/7 + (
c^4*d^3*x^9*(a + b*ArcSinh[c*x]))/3 + (c^6*d^3*x^11*(a + b*ArcSinh[c*x]))/11

________________________________________________________________________________________

Rubi [A]  time = 0.281484, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {270, 5730, 12, 1799, 1620} \[ \frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{11/2}}{121 c^5}+\frac{4 b d^3 \left (c^2 x^2+1\right )^{9/2}}{297 c^5}-\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{1617 c^5}-\frac{2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{1925 c^5}-\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{3465 c^5}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{1155 c^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]

[Out]

(-16*b*d^3*Sqrt[1 + c^2*x^2])/(1155*c^5) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(3465*c^5) - (2*b*d^3*(1 + c^2*x^2)^(
5/2))/(1925*c^5) - (b*d^3*(1 + c^2*x^2)^(7/2))/(1617*c^5) + (4*b*d^3*(1 + c^2*x^2)^(9/2))/(297*c^5) - (b*d^3*(
1 + c^2*x^2)^(11/2))/(121*c^5) + (d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^2*d^3*x^7*(a + b*ArcSinh[c*x]))/7 + (
c^4*d^3*x^9*(a + b*ArcSinh[c*x]))/3 + (c^6*d^3*x^11*(a + b*ArcSinh[c*x]))/11

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^4 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )}{1155 \sqrt{1+c^2 x^2}} \, dx\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (b c d^3\right ) \int \frac{x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )}{\sqrt{1+c^2 x^2}} \, dx}{1155}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (231+495 c^2 x+385 c^4 x^2+105 c^6 x^3\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{2310}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (b c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{16}{c^4 \sqrt{1+c^2 x}}+\frac{8 \sqrt{1+c^2 x}}{c^4}+\frac{6 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1+c^2 x\right )^{5/2}}{c^4}-\frac{140 \left (1+c^2 x\right )^{7/2}}{c^4}+\frac{105 \left (1+c^2 x\right )^{9/2}}{c^4}\right ) \, dx,x,x^2\right )}{2310}\\ &=-\frac{16 b d^3 \sqrt{1+c^2 x^2}}{1155 c^5}-\frac{8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{3465 c^5}-\frac{2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{1925 c^5}-\frac{b d^3 \left (1+c^2 x^2\right )^{7/2}}{1617 c^5}+\frac{4 b d^3 \left (1+c^2 x^2\right )^{9/2}}{297 c^5}-\frac{b d^3 \left (1+c^2 x^2\right )^{11/2}}{121 c^5}+\frac{1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.124513, size = 143, normalized size = 0.63 \[ \frac{d^3 \left (3465 a c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )-b \sqrt{c^2 x^2+1} \left (33075 c^{10} x^{10}+111475 c^8 x^8+117625 c^6 x^6+18933 c^4 x^4-25244 c^2 x^2+50488\right )+3465 b c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right ) \sinh ^{-1}(c x)\right )}{4002075 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]

[Out]

(d^3*(3465*a*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - b*Sqrt[1 + c^2*x^2]*(50488 - 25244*c^2*
x^2 + 18933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10) + 3465*b*c^5*x^5*(231 + 495*c^2*x^2 +
385*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]))/(4002075*c^5)

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 206, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({d}^{3}a \left ({\frac{{c}^{11}{x}^{11}}{11}}+{\frac{{c}^{9}{x}^{9}}{3}}+{\frac{3\,{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{5}{x}^{5}}{5}} \right ) +{d}^{3}b \left ({\frac{{\it Arcsinh} \left ( cx \right ){c}^{11}{x}^{11}}{11}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{9}{x}^{9}}{3}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}}{7}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}}{5}}-{\frac{{c}^{10}{x}^{10}}{121}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{91\,{c}^{8}{x}^{8}}{3267}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4705\,{c}^{6}{x}^{6}}{160083}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{6311\,{c}^{4}{x}^{4}}{1334025}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{25244\,{c}^{2}{x}^{2}}{4002075}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{50488}{4002075}\sqrt{{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x)

[Out]

1/c^5*(d^3*a*(1/11*c^11*x^11+1/3*c^9*x^9+3/7*c^7*x^7+1/5*c^5*x^5)+d^3*b*(1/11*arcsinh(c*x)*c^11*x^11+1/3*arcsi
nh(c*x)*c^9*x^9+3/7*arcsinh(c*x)*c^7*x^7+1/5*arcsinh(c*x)*c^5*x^5-1/121*c^10*x^10*(c^2*x^2+1)^(1/2)-91/3267*c^
8*x^8*(c^2*x^2+1)^(1/2)-4705/160083*c^6*x^6*(c^2*x^2+1)^(1/2)-6311/1334025*c^4*x^4*(c^2*x^2+1)^(1/2)+25244/400
2075*c^2*x^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [B]  time = 1.08555, size = 628, normalized size = 2.78 \begin{align*} \frac{1}{11} \, a c^{6} d^{3} x^{11} + \frac{1}{3} \, a c^{4} d^{3} x^{9} + \frac{3}{7} \, a c^{2} d^{3} x^{7} + \frac{1}{7623} \,{\left (693 \, x^{11} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{63 \, \sqrt{c^{2} x^{2} + 1} x^{10}}{c^{2}} - \frac{70 \, \sqrt{c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac{80 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{6}} - \frac{96 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{10}} - \frac{256 \, \sqrt{c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{945} \,{\left (315 \, x^{9} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac{40 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac{64 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{5} \, a d^{3} x^{5} + \frac{3}{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 )} b c^{2} d^{3} + \frac{1}{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 )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 + 3/7*a*c^2*d^3*x^7 + 1/7623*(693*x^11*arcsinh(c*x) - (63*sqrt(c^2*x^2
+ 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(c^2*x^2 + 1)*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x^4/c^8 +
128*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c)*b*c^6*d^3 + 1/945*(315*x^9*arcsinh(c*x) - (35*
sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)
*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10)*c)*b*c^4*d^3 + 1/5*a*d^3*x^5 + 3/245*(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)*b
*c^2*d^3 + 1/75*(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)*b*d^3

________________________________________________________________________________________

Fricas [A]  time = 2.70114, size = 509, normalized size = 2.25 \begin{align*} \frac{363825 \, a c^{11} d^{3} x^{11} + 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} + 800415 \, a c^{5} d^{3} x^{5} + 3465 \,{\left (105 \, b c^{11} d^{3} x^{11} + 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} + 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (33075 \, b c^{10} d^{3} x^{10} + 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} + 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} + 50488 \, b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{4002075 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/4002075*(363825*a*c^11*d^3*x^11 + 1334025*a*c^9*d^3*x^9 + 1715175*a*c^7*d^3*x^7 + 800415*a*c^5*d^3*x^5 + 346
5*(105*b*c^11*d^3*x^11 + 385*b*c^9*d^3*x^9 + 495*b*c^7*d^3*x^7 + 231*b*c^5*d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1
)) - (33075*b*c^10*d^3*x^10 + 111475*b*c^8*d^3*x^8 + 117625*b*c^6*d^3*x^6 + 18933*b*c^4*d^3*x^4 - 25244*b*c^2*
d^3*x^2 + 50488*b*d^3)*sqrt(c^2*x^2 + 1))/c^5

________________________________________________________________________________________

Sympy [A]  time = 62.2965, size = 289, normalized size = 1.28 \begin{align*} \begin{cases} \frac{a c^{6} d^{3} x^{11}}{11} + \frac{a c^{4} d^{3} x^{9}}{3} + \frac{3 a c^{2} d^{3} x^{7}}{7} + \frac{a d^{3} x^{5}}{5} + \frac{b c^{6} d^{3} x^{11} \operatorname{asinh}{\left (c x \right )}}{11} - \frac{b c^{5} d^{3} x^{10} \sqrt{c^{2} x^{2} + 1}}{121} + \frac{b c^{4} d^{3} x^{9} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{91 b c^{3} d^{3} x^{8} \sqrt{c^{2} x^{2} + 1}}{3267} + \frac{3 b c^{2} d^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{4705 b c d^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{160083} + \frac{b d^{3} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{6311 b d^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{1334025 c} + \frac{25244 b d^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac{50488 b d^{3} \sqrt{c^{2} x^{2} + 1}}{4002075 c^{5}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)

[Out]

Piecewise((a*c**6*d**3*x**11/11 + a*c**4*d**3*x**9/3 + 3*a*c**2*d**3*x**7/7 + a*d**3*x**5/5 + b*c**6*d**3*x**1
1*asinh(c*x)/11 - b*c**5*d**3*x**10*sqrt(c**2*x**2 + 1)/121 + b*c**4*d**3*x**9*asinh(c*x)/3 - 91*b*c**3*d**3*x
**8*sqrt(c**2*x**2 + 1)/3267 + 3*b*c**2*d**3*x**7*asinh(c*x)/7 - 4705*b*c*d**3*x**6*sqrt(c**2*x**2 + 1)/160083
+ b*d**3*x**5*asinh(c*x)/5 - 6311*b*d**3*x**4*sqrt(c**2*x**2 + 1)/(1334025*c) + 25244*b*d**3*x**2*sqrt(c**2*x
**2 + 1)/(4002075*c**3) - 50488*b*d**3*sqrt(c**2*x**2 + 1)/(4002075*c**5), Ne(c, 0)), (a*d**3*x**5/5, True))

________________________________________________________________________________________

Giac [B]  time = 1.82439, size = 574, normalized size = 2.54 \begin{align*} \frac{1}{11} \, a c^{6} d^{3} x^{11} + \frac{1}{3} \, a c^{4} d^{3} x^{9} + \frac{3}{7} \, a c^{2} d^{3} x^{7} + \frac{1}{7623} \,{\left (693 \, x^{11} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{63 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{11}{2}} - 385 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{9}{2}} + 990 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 1386 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 693 \, \sqrt{c^{2} x^{2} + 1}}{c^{11}}\right )} b c^{6} d^{3} + \frac{1}{945} \,{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{9}{2}} - 180 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 420 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} + 1}}{c^{9}}\right )} b c^{4} d^{3} + \frac{1}{5} \, a d^{3} x^{5} + \frac{3}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 21 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{c^{2} x^{2} + 1}}{c^{7}}\right )} b c^{2} d^{3} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 + 3/7*a*c^2*d^3*x^7 + 1/7623*(693*x^11*log(c*x + sqrt(c^2*x^2 + 1)) -
(63*(c^2*x^2 + 1)^(11/2) - 385*(c^2*x^2 + 1)^(9/2) + 990*(c^2*x^2 + 1)^(7/2) - 1386*(c^2*x^2 + 1)^(5/2) + 1155
*(c^2*x^2 + 1)^(3/2) - 693*sqrt(c^2*x^2 + 1))/c^11)*b*c^6*d^3 + 1/945*(315*x^9*log(c*x + sqrt(c^2*x^2 + 1)) -
(35*(c^2*x^2 + 1)^(9/2) - 180*(c^2*x^2 + 1)^(7/2) + 378*(c^2*x^2 + 1)^(5/2) - 420*(c^2*x^2 + 1)^(3/2) + 315*sq
rt(c^2*x^2 + 1))/c^9)*b*c^4*d^3 + 1/5*a*d^3*x^5 + 3/245*(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)*b*c^2*d^3 + 1/75*(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)*
b*d^3