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

Optimal. Leaf size=465 \[ \frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{231} d^3 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {32 b d^3 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {8 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}-\frac {256 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {100976 b^2 d^3 x}{4002075 c^4}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125} \]

[Out]

100976/4002075*b^2*d^3*x/c^4-50488/12006225*b^2*d^3*x^3/c^2+12622/6670125*b^2*d^3*x^5+9410/1120581*b^2*c^2*d^3
*x^7+182/29403*b^2*c^4*d^3*x^9+2/1331*b^2*c^6*d^3*x^11-16/693*b*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c^5+4
/1155*b*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c^5-2/1617*b*d^3*(c^2*x^2+1)^(7/2)*(a+b*arcsinh(c*x))/c^5+8/2
97*b*d^3*(c^2*x^2+1)^(9/2)*(a+b*arcsinh(c*x))/c^5-2/121*b*d^3*(c^2*x^2+1)^(11/2)*(a+b*arcsinh(c*x))/c^5+16/115
5*d^3*x^5*(a+b*arcsinh(c*x))^2+8/231*d^3*x^5*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+2/33*d^3*x^5*(c^2*x^2+1)^2*(a+b*
arcsinh(c*x))^2+1/11*d^3*x^5*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2-256/17325*b*d^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)
^(1/2)/c^5+128/17325*b*d^3*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-32/5775*b*d^3*x^4*(a+b*arcsinh(c*x))*(
c^2*x^2+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 1.06, antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5744, 5661, 5758, 5717, 8, 30, 266, 43, 5732, 12, 1153} \[ \frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{231} d^3 x^5 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {32 b d^3 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}+\frac {128 b d^3 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {8 b d^3 \left (c^2 x^2+1\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}-\frac {256 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {100976 b^2 d^3 x}{4002075 c^4}+\frac {12622 b^2 d^3 x^5}{6670125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) + (12622*b^2*d^3*x^5)/6670125 + (9410*b^
2*c^2*d^3*x^7)/1120581 + (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 - (256*b*d^3*Sqrt[1 + c^2*x^2
]*(a + b*ArcSinh[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(17325*c^3) - (32
*b*d^3*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5775*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x
]))/(693*c^5) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(1155*c^5) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(1617*c^5) + (8*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(297*c^5) - (2*b*d^3*(1 +
c^2*x^2)^(11/2)*(a + b*ArcSinh[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSinh[c*x])^2)/1155 + (8*d^3*x^5*(1 +
c^2*x^2)*(a + b*ArcSinh[c*x])^2)/231 + (2*d^3*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/33 + (d^3*x^5*(1 + c
^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/11

Rule 8

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

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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 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 5732

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(1 + c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSinh[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 +
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2,
0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 5744

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int
[(f*x)^m*(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]
)/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

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]

Rubi steps

\begin {align*} \int x^4 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} (6 d) \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{11} \left (2 b c d^3\right ) \int x^5 \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 )}{77 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{99 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{33} \left (8 d^2\right ) \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{33} \left (4 b c d^3\right ) \int x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{11} \left (2 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (8-28 c^2 x^2+63 c^4 x^4\right )}{693 c^6} \, dx\\ &=-\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{165 c^5}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{231 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{231} \left (16 d^3\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \left (8-28 c^2 x^2+63 c^4 x^4\right ) \, dx}{7623 c^4}-\frac {1}{231} \left (16 b c d^3\right ) \int x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{33} \left (4 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx\\ &=-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (2 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+113 c^6 x^6+161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{7623 c^4}+\frac {\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right ) \, dx}{10395 c^4}-\frac {\left (32 b c d^3\right ) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{1155}+\frac {1}{231} \left (16 b^2 c^2 d^3\right ) \int \frac {8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=\frac {16 b^2 d^3 x}{7623 c^4}-\frac {8 b^2 d^3 x^3}{22869 c^2}+\frac {2 b^2 d^3 x^5}{12705}+\frac {226 b^2 c^2 d^3 x^7}{53361}+\frac {46 b^2 c^4 d^3 x^9}{9801}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (32 b^2 d^3\right ) \int x^4 \, dx}{5775}+\frac {\left (4 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+50 c^6 x^6+35 c^8 x^8\right ) \, dx}{10395 c^4}+\frac {\left (16 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{24255 c^4}+\frac {\left (128 b d^3\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{5775 c}\\ &=\frac {8368 b^2 d^3 x}{800415 c^4}-\frac {4184 b^2 d^3 x^3}{2401245 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (256 b d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{17325 c^3}-\frac {\left (128 b^2 d^3\right ) \int x^2 \, dx}{17325 c^2}\\ &=\frac {8368 b^2 d^3 x}{800415 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (256 b^2 d^3\right ) \int 1 \, dx}{17325 c^4}\\ &=\frac {100976 b^2 d^3 x}{4002075 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} \left (a+b \sinh ^{-1}(c x)\right )}{121 c^5}+\frac {16 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 299, normalized size = 0.64 \[ \frac {d^3 \left (12006225 a^2 c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )-6930 a b \sqrt {c^2 x^2+1} \left (33075 c^{10} x^{10}+111475 c^8 x^8+117625 c^6 x^6+18933 c^4 x^4-25244 c^2 x^2+50488\right )-6930 b \sinh ^{-1}(c x) \left (b \sqrt {c^2 x^2+1} \left (33075 c^{10} x^{10}+111475 c^8 x^8+117625 c^6 x^6+18933 c^4 x^4-25244 c^2 x^2+50488\right )-3465 a c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right )\right )[phone] b^2 c^5 x^5 \left (105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right ) \sinh ^{-1}(c x)^2+2 b^2 c x \left (10418625 c^{10} x^{10}[phone] c^8 x^8+58224375 c^6 x^6+13120569 c^4 x^4-29156820 c^2 x^2+174940920\right )\right )}{13867189875 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(12006225*a^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - 6930*a*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) + 2*b^2*c*x*(174940920 -
29156820*c^2*x^2 + 13120569*c^4*x^4 + 58224375*c^6*x^6 + 42917875*c^8*x^8 + 10418625*c^10*x^10) - 6930*b*(-34
65*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 + 18
933*c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10))*ArcSinh[c*x] + 12006225*b^2*c^5*x^5*(231 + 4
95*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]^2))/(13867189875*c^5)

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fricas [A]  time = 0.60, size = 444, normalized size = 0.95 \[ \frac {10418625 \, {\left (121 \, a^{2} + 2 \, b^{2}\right )} c^{11} d^{3} x^{11} + 471625 \, {\left (9801 \, a^{2} + 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \, {\left (480249 \, a^{2} + 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} + 2079 \, {\left (1334025 \, a^{2} + 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} - 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \, {\left (105 \, b^{2} c^{11} d^{3} x^{11} + 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} + 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6930 \, {\left (363825 \, a b c^{11} d^{3} x^{11} + 1334025 \, a b c^{9} d^{3} x^{9} + 1715175 \, a b c^{7} d^{3} x^{7} + 800415 \, a b c^{5} d^{3} x^{5} - {\left (33075 \, b^{2} c^{10} d^{3} x^{10} + 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} + 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} + 50488 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6930 \, {\left (33075 \, a b c^{10} d^{3} x^{10} + 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} + 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} + 50488 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \]

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))^2,x, algorithm="fricas")

[Out]

1/13867189875*(10418625*(121*a^2 + 2*b^2)*c^11*d^3*x^11 + 471625*(9801*a^2 + 182*b^2)*c^9*d^3*x^9 + 12375*(480
249*a^2 + 9410*b^2)*c^7*d^3*x^7 + 2079*(1334025*a^2 + 12622*b^2)*c^5*d^3*x^5 - 58313640*b^2*c^3*d^3*x^3 + 3498
81840*b^2*c*d^3*x + 12006225*(105*b^2*c^11*d^3*x^11 + 385*b^2*c^9*d^3*x^9 + 495*b^2*c^7*d^3*x^7 + 231*b^2*c^5*
d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6930*(363825*a*b*c^11*d^3*x^11 + 1334025*a*b*c^9*d^3*x^9 + 1715175*a
*b*c^7*d^3*x^7 + 800415*a*b*c^5*d^3*x^5 - (33075*b^2*c^10*d^3*x^10 + 111475*b^2*c^8*d^3*x^8 + 117625*b^2*c^6*d
^3*x^6 + 18933*b^2*c^4*d^3*x^4 - 25244*b^2*c^2*d^3*x^2 + 50488*b^2*d^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*
x^2 + 1)) - 6930*(33075*a*b*c^10*d^3*x^10 + 111475*a*b*c^8*d^3*x^8 + 117625*a*b*c^6*d^3*x^6 + 18933*a*b*c^4*d^
3*x^4 - 25244*a*b*c^2*d^3*x^2 + 50488*a*b*d^3)*sqrt(c^2*x^2 + 1))/c^5

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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))^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.10, size = 520, normalized size = 1.12 \[ \frac {d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{4}}{11}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \arcsinh \left (c x \right )^{2} c x}{1155}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{1155}-\frac {5487704 c x \left (c^{2} x^{2}+1\right )^{2}}{4622396625}-\frac {16 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}-\frac {606416 c x \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {428 c x \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {4 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {148174 c x \left (c^{2} x^{2}+1\right )^{3}}{110937519}+\frac {34 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{5}}{1331}-\frac {32 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{1155}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {382986368 c x}{13867189875}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{11} x^{11}}{11}+\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}\right )}{c^{5}} \]

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))^2,x)

[Out]

1/c^5*(d^3*a^2*(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^2*(1/11*arcsinh(c*x)^2*c^3*x^3*(c^2*
x^2+1)^4-1/33*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^4+16/1155*arcsinh(c*x)^2*c*x+1/231*arcsinh(c*x)^2*c*x*(c^2*x^2+1)
^3+2/385*arcsinh(c*x)^2*c*x*(c^2*x^2+1)^2+8/1155*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-5487704/4622396625*c*x*(c^2*x^
2+1)^2-16/3465*arcsinh(c*x)*(c^2*x^2+1)^(3/2)-606416/13867189875*c*x*(c^2*x^2+1)-428/323433*c*x*(c^2*x^2+1)^4-
2/1617*arcsinh(c*x)*(c^2*x^2+1)^(7/2)-4/1925*arcsinh(c*x)*(c^2*x^2+1)^(5/2)-148174/110937519*c*x*(c^2*x^2+1)^3
+34/3267*arcsinh(c*x)*(c^2*x^2+1)^(9/2)+2/1331*c*x*(c^2*x^2+1)^5-32/1155*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-2/121*
arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(9/2)[phone]/13867189875*c*x)+2*d^3*a*b*(1/11*arcsinh(c*x)*c^11*x^11+1/3*a
rcsinh(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)-91/326
7*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
/4002075*c^2*x^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*x^2+1)^(1/2)))

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maxima [B]  time = 0.47, size = 1109, normalized size = 2.38 \[ \text {result too large to display} \]

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))^2,x, algorithm="maxima")

[Out]

1/11*b^2*c^6*d^3*x^11*arcsinh(c*x)^2 + 1/11*a^2*c^6*d^3*x^11 + 1/3*b^2*c^4*d^3*x^9*arcsinh(c*x)^2 + 1/3*a^2*c^
4*d^3*x^9 + 3/7*b^2*c^2*d^3*x^7*arcsinh(c*x)^2 + 3/7*a^2*c^2*d^3*x^7 + 2/7623*(693*x^11*arcsinh(c*x) - (63*sqr
t(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)*a*b*c^6*d^3 - 2/26413695*(3465*(63*sq
rt(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*arcsinh(c*x) - (19845*c^10*x^11 - 269
50*c^8*x^9 + 39600*c^6*x^7 - 66528*c^4*x^5 + 147840*c^2*x^3 - 887040*x)/c^10)*b^2*c^6*d^3 + 1/5*b^2*d^3*x^5*ar
csinh(c*x)^2 + 2/945*(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)*a*b*c^4*d^3 - 2/297
675*(315*(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*arcsinh(c*x) - (1225*c^8*x^9 - 1800*c^6*x^7 + 3024*c^4*x
^5 - 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^4*d^3 + 1/5*a^2*d^3*x^5 + 6/245*(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
^2*d^3 - 2/8575*(105*(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*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^2*d^
3 + 2/75*(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)*a*b*d^3 - 2/1125*(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*d^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 71.19, size = 702, normalized size = 1.51 \[ \begin {cases} \frac {a^{2} c^{6} d^{3} x^{11}}{11} + \frac {a^{2} c^{4} d^{3} x^{9}}{3} + \frac {3 a^{2} c^{2} d^{3} x^{7}}{7} + \frac {a^{2} d^{3} x^{5}}{5} + \frac {2 a b c^{6} d^{3} x^{11} \operatorname {asinh}{\left (c x \right )}}{11} - \frac {2 a b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1}}{121} + \frac {2 a b c^{4} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {182 a b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{3267} + \frac {6 a b c^{2} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {9410 a b c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{160083} + \frac {2 a b d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {12622 a b d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1334025 c} + \frac {50488 a b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac {100976 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{5}} + \frac {b^{2} c^{6} d^{3} x^{11} \operatorname {asinh}^{2}{\left (c x \right )}}{11} + \frac {2 b^{2} c^{6} d^{3} x^{11}}{1331} - \frac {2 b^{2} c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{121} + \frac {b^{2} c^{4} d^{3} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {182 b^{2} c^{4} d^{3} x^{9}}{29403} - \frac {182 b^{2} c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3267} + \frac {3 b^{2} c^{2} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {9410 b^{2} c^{2} d^{3} x^{7}}{1120581} - \frac {9410 b^{2} c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{160083} + \frac {b^{2} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {12622 b^{2} d^{3} x^{5}}{6670125} - \frac {12622 b^{2} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1334025 c} - \frac {50488 b^{2} d^{3} x^{3}}{12006225 c^{2}} + \frac {50488 b^{2} d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4002075 c^{3}} + \frac {100976 b^{2} d^{3} x}{4002075 c^{4}} - \frac {100976 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \]

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))**2,x)

[Out]

Piecewise((a**2*c**6*d**3*x**11/11 + a**2*c**4*d**3*x**9/3 + 3*a**2*c**2*d**3*x**7/7 + a**2*d**3*x**5/5 + 2*a*
b*c**6*d**3*x**11*asinh(c*x)/11 - 2*a*b*c**5*d**3*x**10*sqrt(c**2*x**2 + 1)/121 + 2*a*b*c**4*d**3*x**9*asinh(c
*x)/3 - 182*a*b*c**3*d**3*x**8*sqrt(c**2*x**2 + 1)/3267 + 6*a*b*c**2*d**3*x**7*asinh(c*x)/7 - 9410*a*b*c*d**3*
x**6*sqrt(c**2*x**2 + 1)/160083 + 2*a*b*d**3*x**5*asinh(c*x)/5 - 12622*a*b*d**3*x**4*sqrt(c**2*x**2 + 1)/(1334
025*c) + 50488*a*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(4002075*c**3) - 100976*a*b*d**3*sqrt(c**2*x**2 + 1)/(4002075
*c**5) + b**2*c**6*d**3*x**11*asinh(c*x)**2/11 + 2*b**2*c**6*d**3*x**11/1331 - 2*b**2*c**5*d**3*x**10*sqrt(c**
2*x**2 + 1)*asinh(c*x)/121 + b**2*c**4*d**3*x**9*asinh(c*x)**2/3 + 182*b**2*c**4*d**3*x**9/29403 - 182*b**2*c*
*3*d**3*x**8*sqrt(c**2*x**2 + 1)*asinh(c*x)/3267 + 3*b**2*c**2*d**3*x**7*asinh(c*x)**2/7 + 9410*b**2*c**2*d**3
*x**7/1120581 - 9410*b**2*c*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/160083 + b**2*d**3*x**5*asinh(c*x)**2/5 +
12622*b**2*d**3*x**5/6670125 - 12622*b**2*d**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(1334025*c) - 50488*b**2*d
**3*x**3/(12006225*c**2) + 50488*b**2*d**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4002075*c**3) + 100976*b**2*d*
*3*x/(4002075*c**4) - 100976*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4002075*c**5), Ne(c, 0)), (a**2*d**3*x*
*5/5, True))

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