3.1.43 \(\int (f+g x)^3 (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x)) \, dx\) [43]

Optimal. Leaf size=1228 \[ -\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

15/64*d^2*f*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+5/24*d^2*f^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*
d*x^2+d)^(1/2)+1/6*d^2*f^3*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)-1/7*d^2*g^3*(c^2*x^2+1)^3*(a
+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4+1/9*d^2*g^3*(c^2*x^2+1)^4*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4-
1/36*b*d^2*f^3*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/c-3/7*b*d^2*f^2*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/
2)-15/256*b*d^2*f*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-3/7*b*c*d^2*f^2*g*x^3*(c^2*d*x^2+d)^(1/2)/(c
^2*x^2+1)^(1/2)-59/256*b*c*d^2*f*g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-9/35*b*c^3*d^2*f^2*g*x^5*(c^2*d
*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-17/96*b*c^3*d^2*f*g^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/49*b*c^5*d^2
*f^2*g*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/64*b*c^5*d^2*f*g^2*x^8*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2
)-15/256*d^2*f*g^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)+5/16*d^2*f^3*x*(a+b*arcsin
h(c*x))*(c^2*d*x^2+d)^(1/2)-25/96*b*c*d^2*f^3*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/189*b*d^2*g^3*x^3*(c
^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/96*b*c^3*d^2*f^3*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/21*b*c*d^
2*g^3*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-19/441*b*c^3*d^2*g^3*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)
-1/81*b*c^5*d^2*g^3*x^9*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/32*d^2*f^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^
(1/2)/b/c/(c^2*x^2+1)^(1/2)+15/128*d^2*f*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+5/16*d^2*f*g^2*x^3*(
c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+3/8*d^2*f*g^2*x^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^
2+d)^(1/2)+3/7*d^2*f^2*g*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+2/63*b*d^2*g^3*x*(c^2*d*x^2+
d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.75, antiderivative size = 1228, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 267, 5798, 200, 5808, 5806, 5812, 272, 45, 5804, 12, 380} \begin {gather*} -\frac {b c^5 d^2 g^3 \sqrt {c^2 d x^2+d} x^9}{81 \sqrt {c^2 x^2+1}}-\frac {3 b c^5 d^2 f g^2 \sqrt {c^2 d x^2+d} x^8}{64 \sqrt {c^2 x^2+1}}-\frac {19 b c^3 d^2 g^3 \sqrt {c^2 d x^2+d} x^7}{441 \sqrt {c^2 x^2+1}}-\frac {3 b c^5 d^2 f^2 g \sqrt {c^2 d x^2+d} x^7}{49 \sqrt {c^2 x^2+1}}-\frac {17 b c^3 d^2 f g^2 \sqrt {c^2 d x^2+d} x^6}{96 \sqrt {c^2 x^2+1}}-\frac {b c d^2 g^3 \sqrt {c^2 d x^2+d} x^5}{21 \sqrt {c^2 x^2+1}}-\frac {9 b c^3 d^2 f^2 g \sqrt {c^2 d x^2+d} x^5}{35 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 f^3 \sqrt {c^2 d x^2+d} x^4}{96 \sqrt {c^2 x^2+1}}-\frac {59 b c d^2 f g^2 \sqrt {c^2 d x^2+d} x^4}{256 \sqrt {c^2 x^2+1}}+\frac {15}{64} d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac {3}{8} d^2 f g^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3-\frac {b d^2 g^3 \sqrt {c^2 d x^2+d} x^3}{189 c \sqrt {c^2 x^2+1}}-\frac {3 b c d^2 f^2 g \sqrt {c^2 d x^2+d} x^3}{7 \sqrt {c^2 x^2+1}}-\frac {25 b c d^2 f^3 \sqrt {c^2 d x^2+d} x^2}{96 \sqrt {c^2 x^2+1}}-\frac {15 b d^2 f g^2 \sqrt {c^2 d x^2+d} x^2}{256 c \sqrt {c^2 x^2+1}}+\frac {5}{16} d^2 f^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {2 b d^2 g^3 \sqrt {c^2 d x^2+d} x}{63 c^3 \sqrt {c^2 x^2+1}}-\frac {3 b d^2 f^2 g \sqrt {c^2 d x^2+d} x}{7 c \sqrt {c^2 x^2+1}}+\frac {5 d^2 f^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c^2 x^2+1}}+\frac {d^2 g^3 \left (c^2 x^2+1\right )^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac {d^2 g^3 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {3 d^2 f^2 g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {b d^2 f^3 \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-3*b*d^2*f^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) + (2*b*d^2*g^3*x*Sqrt[d + c^2*d*x^2])/(63*c^3*S
qrt[1 + c^2*x^2]) - (25*b*c*d^2*f^3*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt
[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) -
(b*d^2*g^3*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d + c^2*d*x^2])/(96
*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(256*Sqrt[1 + c^2*x^2]) - (9*b*c^3*d^2*f^2*g*
x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2])
- (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d + c^2*d
*x^2])/(49*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (3*b*c^5*
d^2*f*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1
+ c^2*x^2]) - (b*d^2*f^3*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^2*f^3*x*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/16 + (15*d^2*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (15*d^2*f*g^2*
x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/64 + (5*d^2*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x]))/24 + (5*d^2*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (d^2*f^3*x*(1 + c
^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (3*d^2*f*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/8 + (3*d^2*f^2*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) - (d^2
*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4) + (d^2*g^3*(1 + c^2*x^2)^4*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x]))/(9*c^4) + (5*d^2*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1
+ c^2*x^2]) - (15*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt[1 + c^2*x^2])

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 45

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 200

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]

Rule 267

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 272

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 380

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

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5785

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

Rule 5786

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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 5804

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

Rule 5806

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

Rule 5808

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^
p/(1 + c^2*x^2)^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]

Rule 5812

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

Rule 5838

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

Rule 5845

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

Rubi steps

\begin {align*} \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f^2 g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f g^2 x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g^3 x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (x+2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (15 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (15 b d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1+c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 5.02, size = 1667, normalized size = 1.36 \begin {gather*} \frac {3 a d^2 f^2 g \sqrt {d+c^2 d x^2}}{7 c^2}-\frac {2 a d^2 g^3 \sqrt {d+c^2 d x^2}}{63 c^4}+\frac {11}{16} a d^2 f^3 x \sqrt {d+c^2 d x^2}+\frac {15 a d^2 f g^2 x \sqrt {d+c^2 d x^2}}{128 c^2}+\frac {9}{7} a d^2 f^2 g x^2 \sqrt {d+c^2 d x^2}+\frac {a d^2 g^3 x^2 \sqrt {d+c^2 d x^2}}{63 c^2}+\frac {13}{24} a c^2 d^2 f^3 x^3 \sqrt {d+c^2 d x^2}+\frac {59}{64} a d^2 f g^2 x^3 \sqrt {d+c^2 d x^2}+\frac {9}{7} a c^2 d^2 f^2 g x^4 \sqrt {d+c^2 d x^2}+\frac {5}{21} a d^2 g^3 x^4 \sqrt {d+c^2 d x^2}+\frac {1}{6} a c^4 d^2 f^3 x^5 \sqrt {d+c^2 d x^2}+\frac {17}{16} a c^2 d^2 f g^2 x^5 \sqrt {d+c^2 d x^2}+\frac {3}{7} a c^4 d^2 f^2 g x^6 \sqrt {d+c^2 d x^2}+\frac {19}{63} a c^2 d^2 g^3 x^6 \sqrt {d+c^2 d x^2}+\frac {3}{8} a c^4 d^2 f g^2 x^7 \sqrt {d+c^2 d x^2}+\frac {1}{9} a c^4 d^2 g^3 x^8 \sqrt {d+c^2 d x^2}-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}+\frac {5 b d^2 f \left (8 c^2 f^2-3 g^2\right ) \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{256 c^3 \sqrt {1+c^2 x^2}}-\frac {3 b d^2 f \left (5 c^2 f^2-g^2\right ) \sqrt {d+c^2 d x^2} \cosh \left (2 \sinh ^{-1}(c x)\right )}{128 c^3 \sqrt {1+c^2 x^2}}-\frac {3 b d^2 f^3 \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b d^2 f g^2 \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )}{512 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )}{1152 c \sqrt {1+c^2 x^2}}-\frac {b d^2 f g^2 \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )}{384 c^3 \sqrt {1+c^2 x^2}}-\frac {3 b d^2 f g^2 \sqrt {d+c^2 d x^2} \cosh \left (8 \sinh ^{-1}(c x)\right )}{8192 c^3 \sqrt {1+c^2 x^2}}+\frac {5 a d^{5/2} f^3 \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{16 c}-\frac {15 a d^{5/2} f g^2 \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{128 c^3}+\frac {b d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (27648 c^2 f^2 g \sqrt {1+c^2 x^2}-2048 g^3 \sqrt {1+c^2 x^2}+82944 c^4 f^2 g x^2 \sqrt {1+c^2 x^2}+1024 c^2 g^3 x^2 \sqrt {1+c^2 x^2}+82944 c^6 f^2 g x^4 \sqrt {1+c^2 x^2}+15360 c^4 g^3 x^4 \sqrt {1+c^2 x^2}+27648 c^8 f^2 g x^6 \sqrt {1+c^2 x^2}+19456 c^6 g^3 x^6 \sqrt {1+c^2 x^2}+7168 c^8 g^3 x^8 \sqrt {1+c^2 x^2}+3024 c f \left (5 c^2 f^2-g^2\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )+1512 c f \left (2 c^2 f^2+g^2\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+336 c^3 f^3 \sinh \left (6 \sinh ^{-1}(c x)\right )+1008 c f g^2 \sinh \left (6 \sinh ^{-1}(c x)\right )+189 c f g^2 \sinh \left (8 \sinh ^{-1}(c x)\right )\right )}{64512 c^4 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(3*a*d^2*f^2*g*Sqrt[d + c^2*d*x^2])/(7*c^2) - (2*a*d^2*g^3*Sqrt[d + c^2*d*x^2])/(63*c^4) + (11*a*d^2*f^3*x*Sqr
t[d + c^2*d*x^2])/16 + (15*a*d^2*f*g^2*x*Sqrt[d + c^2*d*x^2])/(128*c^2) + (9*a*d^2*f^2*g*x^2*Sqrt[d + c^2*d*x^
2])/7 + (a*d^2*g^3*x^2*Sqrt[d + c^2*d*x^2])/(63*c^2) + (13*a*c^2*d^2*f^3*x^3*Sqrt[d + c^2*d*x^2])/24 + (59*a*d
^2*f*g^2*x^3*Sqrt[d + c^2*d*x^2])/64 + (9*a*c^2*d^2*f^2*g*x^4*Sqrt[d + c^2*d*x^2])/7 + (5*a*d^2*g^3*x^4*Sqrt[d
+ c^2*d*x^2])/21 + (a*c^4*d^2*f^3*x^5*Sqrt[d + c^2*d*x^2])/6 + (17*a*c^2*d^2*f*g^2*x^5*Sqrt[d + c^2*d*x^2])/1
6 + (3*a*c^4*d^2*f^2*g*x^6*Sqrt[d + c^2*d*x^2])/7 + (19*a*c^2*d^2*g^3*x^6*Sqrt[d + c^2*d*x^2])/63 + (3*a*c^4*d
^2*f*g^2*x^7*Sqrt[d + c^2*d*x^2])/8 + (a*c^4*d^2*g^3*x^8*Sqrt[d + c^2*d*x^2])/9 - (3*b*d^2*f^2*g*x*Sqrt[d + c^
2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) + (2*b*d^2*g^3*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (3*b*c*d^
2*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (b*d^2*g^3*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c
^2*x^2]) - (9*b*c^3*d^2*f^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d + c^2*
d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (19*b*c^
3*d^2*g^3*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[
1 + c^2*x^2]) + (5*b*d^2*f*(8*c^2*f^2 - 3*g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(256*c^3*Sqrt[1 + c^2*x^2])
- (3*b*d^2*f*(5*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]])/(128*c^3*Sqrt[1 + c^2*x^2]) - (3*b*d
^2*f^3*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]])/(256*c*Sqrt[1 + c^2*x^2]) - (3*b*d^2*f*g^2*Sqrt[d + c^2*d*x^2
]*Cosh[4*ArcSinh[c*x]])/(512*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*f^3*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]])/(11
52*c*Sqrt[1 + c^2*x^2]) - (b*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]])/(384*c^3*Sqrt[1 + c^2*x^2]) -
(3*b*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]])/(8192*c^3*Sqrt[1 + c^2*x^2]) + (5*a*d^(5/2)*f^3*Log[
c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(16*c) - (15*a*d^(5/2)*f*g^2*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(
128*c^3) + (b*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(27648*c^2*f^2*g*Sqrt[1 + c^2*x^2] - 2048*g^3*Sqrt[1 + c^2*
x^2] + 82944*c^4*f^2*g*x^2*Sqrt[1 + c^2*x^2] + 1024*c^2*g^3*x^2*Sqrt[1 + c^2*x^2] + 82944*c^6*f^2*g*x^4*Sqrt[1
+ c^2*x^2] + 15360*c^4*g^3*x^4*Sqrt[1 + c^2*x^2] + 27648*c^8*f^2*g*x^6*Sqrt[1 + c^2*x^2] + 19456*c^6*g^3*x^6*
Sqrt[1 + c^2*x^2] + 7168*c^8*g^3*x^8*Sqrt[1 + c^2*x^2] + 3024*c*f*(5*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 151
2*c*f*(2*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + 336*c^3*f^3*Sinh[6*ArcSinh[c*x]] + 1008*c*f*g^2*Sinh[6*ArcSinh[
c*x]] + 189*c*f*g^2*Sinh[8*ArcSinh[c*x]]))/(64512*c^4*Sqrt[1 + c^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2915\) vs. \(2(1080)=2160\).
time = 4.57, size = 2916, normalized size = 2.37

method	result	size
default	\(\text {Expression too large to display}\)	\(2916\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/9*a*g^3*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-2/63*a*g^3/d/c^4*(c^2*d*x^2+d)^(7/2)+3/8*a*f*g^2*x*(c^2*d*x^2+d)^(7/2)
/c^2/d-1/16*a*f*g^2/c^2*x*(c^2*d*x^2+d)^(5/2)-5/64*a*f*g^2/c^2*d*x*(c^2*d*x^2+d)^(3/2)-15/128*a*f*g^2/c^2*d^2*
x*(c^2*d*x^2+d)^(1/2)-15/128*a*f*g^2/c^2*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+3/7*a
*f^2*g*(c^2*d*x^2+d)^(7/2)/c^2/d+1/6*a*f^3*x*(c^2*d*x^2+d)^(5/2)+5/24*a*f^3*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a*f^3
*d^2*x*(c^2*d*x^2+d)^(1/2)+5/16*a*f^3*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(5/256
*(d*(c^2*x^2+1))^(1/2)*f*arcsinh(c*x)^2*(8*c^2*f^2-3*g^2)*d^2/(c^2*x^2+1)^(1/2)/c^3+1/41472*(d*(c^2*x^2+1))^(1
/2)*(256*x^10*c^10+256*(c^2*x^2+1)^(1/2)*x^9*c^9+704*c^8*x^8+576*(c^2*x^2+1)^(1/2)*x^7*c^7+688*x^6*c^6+432*(c^
2*x^2+1)^(1/2)*x^5*c^5+280*c^4*x^4+120*(c^2*x^2+1)^(1/2)*x^3*c^3+41*c^2*x^2+9*(c^2*x^2+1)^(1/2)*c*x+1)*g^3*(-1
+9*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/16384*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x^9+128*(c^2*x^2+1)^(1/2)*x^8*c^8+
320*c^7*x^7+256*(c^2*x^2+1)^(1/2)*c^6*x^6+272*c^5*x^5+160*(c^2*x^2+1)^(1/2)*c^4*x^4+88*c^3*x^3+32*c^2*x^2*(c^2
*x^2+1)^(1/2)+8*c*x+(c^2*x^2+1)^(1/2))*f*g^2*(-1+8*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(
1/2)*(64*c^8*x^8+64*(c^2*x^2+1)^(1/2)*x^7*c^7+144*x^6*c^6+112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4+56*(c^2*x^
2+1)^(1/2)*x^3*c^3+25*c^2*x^2+7*(c^2*x^2+1)^(1/2)*c*x+1)*g*(28*arcsinh(c*x)*c^2*f^2-4*c^2*f^2+7*arcsinh(c*x)*g
^2-g^2)*d^2/c^4/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*(c^2*x^2+1)^(1/2)*c^6*x^6+64*c^5*x^5+4
8*(c^2*x^2+1)^(1/2)*c^4*x^4+38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*f*(6*arcsinh(c*x)
*c^2*f^2-c^2*f^2+18*arcsinh(c*x)*g^2-3*g^2)*d^2/c^3/(c^2*x^2+1)+3/640*(d*(c^2*x^2+1))^(1/2)*(16*x^6*c^6+16*(c^
2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4+20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*x^2+5*(c^2*x^2+1)^(1/2)*c*x+1)*f^2*g*(-1
+5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*(c^2*x^2+1)^(1/2)*c^4*x^4+12*c^
3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*f*(8*arcsinh(c*x)*c^2*f^2-2*c^2*f^2+4*arcsinh(c*x)*
g^2-g^2)*d^2/c^3/(c^2*x^2+1)+1/1152*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2+3*(
c^2*x^2+1)^(1/2)*c*x+1)*g*(81*arcsinh(c*x)*c^2*f^2-27*c^2*f^2-6*arcsinh(c*x)*g^2+2*g^2)*d^2/c^4/(c^2*x^2+1)+3/
256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*f*(10*arcsinh(c*x)*c
^2*f^2-5*c^2*f^2-2*arcsinh(c*x)*g^2+g^2)*d^2/c^3/(c^2*x^2+1)+3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^
(1/2)*c*x+1)*g*(10*arcsinh(c*x)*c^2*f^2-10*c^2*f^2-arcsinh(c*x)*g^2+g^2)*d^2/c^4/(c^2*x^2+1)+3/256*(d*(c^2*x^2
+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)*g*(10*arcsinh(c*x)*c^2*f^2+10*c^2*f^2-arcsinh(c*x)*g^2-g^2)*d^2/c
^4/(c^2*x^2+1)+3/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*f*(
10*arcsinh(c*x)*c^2*f^2+5*c^2*f^2-2*arcsinh(c*x)*g^2-g^2)*d^2/c^3/(c^2*x^2+1)+1/1152*(d*(c^2*x^2+1))^(1/2)*(4*
c^4*x^4-4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*c*x+1)*g*(81*arcsinh(c*x)*c^2*f^2+27*c^2*f^2
-6*arcsinh(c*x)*g^2-2*g^2)*d^2/c^4/(c^2*x^2+1)+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*(c^2*x^2+1)^(1/2)*c^4
*x^4+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*f*(8*arcsinh(c*x)*c^2*f^2+2*c^2*f^2+4*arc
sinh(c*x)*g^2+g^2)*d^2/c^3/(c^2*x^2+1)+3/640*(d*(c^2*x^2+1))^(1/2)*(16*x^6*c^6-16*(c^2*x^2+1)^(1/2)*x^5*c^5+28
*c^4*x^4-20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*x^2-5*(c^2*x^2+1)^(1/2)*c*x+1)*f^2*g*(1+5*arcsinh(c*x))*d^2/c^2/(
c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*(c^2*x^2+1)^(1/2)*c^6*x^6+64*c^5*x^5-48*(c^2*x^2+1)^(1/
2)*c^4*x^4+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*f*(6*arcsinh(c*x)*c^2*f^2+c^2*f^2+
18*arcsinh(c*x)*g^2+3*g^2)*d^2/c^3/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8-64*(c^2*x^2+1)^(1/2)*
x^7*c^7+144*x^6*c^6-112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4-56*(c^2*x^2+1)^(1/2)*x^3*c^3+25*c^2*x^2-7*(c^2*x
^2+1)^(1/2)*c*x+1)*g*(28*arcsinh(c*x)*c^2*f^2+4*c^2*f^2+7*arcsinh(c*x)*g^2+g^2)*d^2/c^4/(c^2*x^2+1)+3/16384*(d
*(c^2*x^2+1))^(1/2)*(128*c^9*x^9-128*(c^2*x^2+1)^(1/2)*x^8*c^8+320*c^7*x^7-256*(c^2*x^2+1)^(1/2)*c^6*x^6+272*c
^5*x^5-160*(c^2*x^2+1)^(1/2)*c^4*x^4+88*c^3*x^3-32*c^2*x^2*(c^2*x^2+1)^(1/2)+8*c*x-(c^2*x^2+1)^(1/2))*f*g^2*(1
+8*arcsinh(c*x))*d^2/c^3/(c^2*x^2+1)+1/41472*(d*(c^2*x^2+1))^(1/2)*(256*x^10*c^10-256*(c^2*x^2+1)^(1/2)*x^9*c^
9+704*c^8*x^8-576*(c^2*x^2+1)^(1/2)*x^7*c^7+688*x^6*c^6-432*(c^2*x^2+1)^(1/2)*x^5*c^5+280*c^4*x^4-120*(c^2*x^2
+1)^(1/2)*x^3*c^3+41*c^2*x^2-9*(c^2*x^2+1)^(1/2)*c*x+1)*g^3*(1+9*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a*d^2*f^3 + (3*a*c^4*d^2*f^2*g + 2*a*c
^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 + 6*a*c^2*d^2*f*g^2)*x^4 + (6*a*c^2*d^2*f^2*g + a*d^2*g^3)*x^3 + (2*a*c^2*d^2
*f^3 + 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^
4*d^2*f^2*g + 2*b*c^2*d^2*g^3)*x^5 + (b*c^4*d^2*f^3 + 6*b*c^2*d^2*f*g^2)*x^4 + (6*b*c^2*d^2*f^2*g + b*d^2*g^3)
*x^3 + (2*b*c^2*d^2*f^3 + 3*b*d^2*f*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3060 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)

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