3.1.40 \(\int (f+g x)^2 (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x)) \, dx\) [40]

Optimal. Leaf size=651 \[ -\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {d g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

3/8*d*f^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/16*d*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+1/8
*d*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/4*d*f^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/
2)+1/6*d*g^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/5*d*f*g*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))
*(c^2*d*x^2+d)^(1/2)/c^2-2/5*b*d*f*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d*f^2*x^2*(c^2*d*x^2+d
)^(1/2)/(c^2*x^2+1)^(1/2)-1/32*b*d*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-4/15*b*c*d*f*g*x^3*(c^2*d*x
^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-7/96*b*c*d*g^2*x^4*(c
^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/25*b*c^3*d*f*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/36*b*c^3*d*g^
2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+3/16*d*f^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1
)^(1/2)-1/32*d*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)

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 651, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 5798, 200, 5808, 5806, 5812} \begin {gather*} \frac {3}{8} d f^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d f^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {c^2 x^2+1}}+\frac {2 d f g \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {d g^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d g^2 x^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {c^2 x^2+1}}-\frac {5 b c d f^2 x^2 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {2 b d f g x \sqrt {c^2 d x^2+d}}{5 c \sqrt {c^2 x^2+1}}-\frac {4 b c d f g x^3 \sqrt {c^2 d x^2+d}}{15 \sqrt {c^2 x^2+1}}-\frac {b d g^2 x^2 \sqrt {c^2 d x^2+d}}{32 c \sqrt {c^2 x^2+1}}-\frac {7 b c d g^2 x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b c^3 d f^2 x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d f g x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}}-\frac {b c^3 d g^2 x^6 \sqrt {c^2 d x^2+d}}{36 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*d*f*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) - (5*b*c*d*f^2*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1
+ c^2*x^2]) - (b*d*g^2*x^2*Sqrt[d + c^2*d*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (4*b*c*d*f*g*x^3*Sqrt[d + c^2*d*x^2
])/(15*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (7*b*c*d*g^2*x^4*Sq
rt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*f*g*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) - (
b*c^3*d*g^2*x^6*Sqrt[d + c^2*d*x^2])/(36*Sqrt[1 + c^2*x^2]) + (3*d*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/8 + (d*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (d*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*A
rcSinh[c*x]))/8 + (d*f^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/4 + (d*g^2*x^3*(1 + c^2*x^2
)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (2*d*f*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/(5*c^2) + (3*d*f^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2]) - (d*g^2*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])

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 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 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 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)^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \left (f^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+2 f g x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+g^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d f^2 \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}{\sqrt {1+c^2 x^2}}+\frac {\left (2 d f g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d 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}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} d f^2 x \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 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 {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (d 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}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \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 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 {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (d 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}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \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 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 {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {\left (d 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}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {d g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d f^2 x \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 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 {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {d g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.46, size = 546, normalized size = 0.84 \begin {gather*} \frac {240 a c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (96 f g \left (1+c^2 x^2\right )^2+30 c^2 f^2 x \left (5+2 c^2 x^2\right )+5 g^2 x \left (3+14 c^2 x^2+8 c^4 x^4\right )\right )-12800 b c d f g \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)\right )-512 b c d f g \sqrt {d+c^2 d x^2} \left (c x \left (-30+5 c^2 x^2+9 c^4 x^4\right )-15 \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right ) \sinh ^{-1}(c x)\right )+3600 a d^{3/2} \left (6 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-7200 b c^2 d f^2 \sqrt {d+c^2 d x^2} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-450 b c^2 d f^2 \sqrt {d+c^2 d x^2} \left (8 \sinh ^{-1}(c x)^2+\cosh \left (4 \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-450 b d g^2 \sqrt {d+c^2 d x^2} \left (8 \sinh ^{-1}(c x)^2+\cosh \left (4 \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )\right )+25 b d g^2 \sqrt {d+c^2 d x^2} \left (72 \sinh ^{-1}(c x)^2+18 \cosh \left (2 \sinh ^{-1}(c x)\right )+9 \cosh \left (4 \sinh ^{-1}(c x)\right )-2 \cosh \left (6 \sinh ^{-1}(c x)\right )+12 \sinh ^{-1}(c x) \left (-3 \sinh \left (2 \sinh ^{-1}(c x)\right )-3 \sinh \left (4 \sinh ^{-1}(c x)\right )+\sinh \left (6 \sinh ^{-1}(c x)\right )\right )\right )}{57600 c^3 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(240*a*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(96*f*g*(1 + c^2*x^2)^2 + 30*c^2*f^2*x*(5 + 2*c^2*x^2) + 5*g^
2*x*(3 + 14*c^2*x^2 + 8*c^4*x^4)) - 12800*b*c*d*f*g*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/
2)*ArcSinh[c*x]) - 512*b*c*d*f*g*Sqrt[d + c^2*d*x^2]*(c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4) - 15*Sqrt[1 + c^2*x^2]
*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]) + 3600*a*d^(3/2)*(6*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqr
t[d]*Sqrt[d + c^2*d*x^2]] - 7200*b*c^2*d*f^2*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSi
nh[c*x] + Sinh[2*ArcSinh[c*x]])) - 450*b*c^2*d*f^2*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]
] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) - 450*b*d*g^2*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[
c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) + 25*b*d*g^2*Sqrt[d + c^2*d*x^2]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*A
rcSinh[c*x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*
Sinh[4*ArcSinh[c*x]] + Sinh[6*ArcSinh[c*x]])))/(57600*c^3*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. \(1584\) vs. \(2(567)=1134\).
time = 5.72, size = 1585, normalized size = 2.43

method	result	size
default	\(\text {Expression too large to display}\)	\(1585\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*g^2*x*(c^2*d*x^2+d)^(5/2)/c^2/d-1/24*a*g^2/c^2*x*(c^2*d*x^2+d)^(3/2)-1/16*a*g^2/c^2*d*x*(c^2*d*x^2+d)^(1
/2)-1/16*a*g^2/c^2*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+2/5*a*f*g/c^2/d*(c^2*d*x^2+
d)^(5/2)+1/4*a*f^2*x*(c^2*d*x^2+d)^(3/2)+3/8*a*f^2*d*x*(c^2*d*x^2+d)^(1/2)+3/8*a*f^2*d^2*ln(x*c^2*d/(c^2*d)^(1
/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(1/32*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*(6*c^2*f^2-g^2)*d/(c^2*x^2
+1)^(1/2)/c^3+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))*g^2*(-1+6*arcsinh(c*x))*d/c^3/(
c^2*x^2+1)+1/400*(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*g*(-1+5*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/512*(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))
*(8*arcsinh(c*x)*c^2*f^2-2*c^2*f^2+4*arcsinh(c*x)*g^2-g^2)*d/c^3/(c^2*x^2+1)+1/48*(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)*f*g*(-1+3*arcsinh(c*x))*d/c^2/(c^2*x^2+1
)+1/256*(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))*(32*arcsinh(c*x)
*c^2*f^2-16*c^2*f^2-2*arcsinh(c*x)*g^2+g^2)*d/c^3/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(
1/2)*c*x+1)*f*g*(arcsinh(c*x)-1)*d/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)
*f*g*(arcsinh(c*x)+1)*d/c^2/(c^2*x^2+1)+1/256*(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))*(32*arcsinh(c*x)*c^2*f^2+16*c^2*f^2-2*arcsinh(c*x)*g^2-g^2)*d/c^3/(c^2*x^2+1)+1/48*(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)*f*g*(1+3*arcsinh(
c*x))*d/c^2/(c^2*x^2+1)+1/512*(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))*(8*arcsinh(c*x)*c^2*f^2+2*c^2*f^2+4*arcsinh(c*x)*g^2+g^2)*d/c^3/(
c^2*x^2+1)+1/400*(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*g*(1+5*arcsinh(c*x))*d/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))*g^2*(1+6*arcsinh(c*x))*d/c^3/(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)^2*(c^2*d*x^2+d)^(3/2)*(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)^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))*(f + g*x)**2, x)

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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)^2*(c^2*d*x^2+d)^(3/2)*(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 )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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