3.3.77 \(\int (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [277]

Optimal. Leaf size=420 \[ \frac {245 b^2 d^2 x \sqrt {d+c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {115 b^2 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt {1+c^2 x^2}} \]

[Out]

5/24*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/6*x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2+245/1152*b^2*
d^2*x*(c^2*d*x^2+d)^(1/2)+65/1728*b^2*d^2*x*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)+1/108*b^2*d^2*x*(c^2*x^2+1)^2*(c^2
*d*x^2+d)^(1/2)-5/48*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c-1/18*b*d^2*(c^2*x^2+1)^(
5/2)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c+5/16*d^2*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)-115/1152*b^2
*d^2*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d^2*x^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1
/2)/(c^2*x^2+1)^(1/2)+5/48*d^2*(a+b*arcsinh(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5786, 5785, 5783, 5776, 327, 221, 5798, 201} \begin {gather*} \frac {5 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt {c^2 x^2+1}}+\frac {5}{16} d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d^2 \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {5 b d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {5 b c d^2 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt {c^2 x^2+1}}+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{108} b^2 d^2 x \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}+\frac {245 b^2 d^2 x \sqrt {c^2 d x^2+d}}{1152}+\frac {65 b^2 d^2 x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{1728}-\frac {115 b^2 d^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(245*b^2*d^2*x*Sqrt[d + c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/1728 + (b^2*d^2*x*
(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/108 - (115*b^2*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(1152*c*Sqrt[1 + c^2
*x^2]) - (5*b*c*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*Sqrt[1 + c^2*x^2]) - (5*b*d^2*(1 + c^2*x
^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(48*c) - (b*d^2*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2]*(a
+ b*ArcSinh[c*x]))/(18*c) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/16 + (5*d*x*(d + c^2*d*x^2)^
(3/2)*(a + b*ArcSinh[c*x])^2)/24 + (x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 + (5*d^2*Sqrt[d + c^2*d*
x^2]*(a + b*ArcSinh[c*x])^3)/(48*b*c*Sqrt[1 + c^2*x^2])

Rule 201

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

Rule 221

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

Rule 327

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

Rule 5776

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 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]

Rubi steps

\begin {align*} \int \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} (5 d) \int \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} \left (5 d^2\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{18 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{12 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{48 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt {1+c^2 x^2}}\\ &=\frac {65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{144 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{64 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}\\ &=\frac {245 b^2 d^2 x \sqrt {d+c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{288 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{128 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 \sqrt {1+c^2 x^2}}\\ &=\frac {245 b^2 d^2 x \sqrt {d+c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {115 b^2 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{1152 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.04, size = 499, normalized size = 1.19 \begin {gather*} \frac {d^2 \left (9504 a^2 c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+7488 a^2 c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+1440 b^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^3-3240 a b \sqrt {d+c^2 d x^2} \cosh \left (2 \sinh ^{-1}(c x)\right )-324 a b \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )+4320 a^2 \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+1620 b^2 \sqrt {d+c^2 d x^2} \sinh \left (2 \sinh ^{-1}(c x)\right )+81 b^2 \sqrt {d+c^2 d x^2} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 \sqrt {d+c^2 d x^2} \sinh \left (6 \sinh ^{-1}(c x)\right )-12 b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (270 b \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b \cosh \left (4 \sinh ^{-1}(c x)\right )+2 b \cosh \left (6 \sinh ^{-1}(c x)\right )-540 a \sinh \left (2 \sinh ^{-1}(c x)\right )-108 a \sinh \left (4 \sinh ^{-1}(c x)\right )-12 a \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+72 b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2 \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )\right )}{13824 c \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(9504*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 7488*a^2*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x
^2] + 2304*a^2*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 1440*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 3
240*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 324*a*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*S
qrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] + 4320*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*
d*x^2]] + 1620*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 81*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]]
+ 4*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] - 12*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh
[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*ArcSinh[c*x]] - 108*a*Sinh[4*ArcS
inh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c
*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]])))/(13824*c*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. \(1567\) vs. \(2(366)=732\).
time = 1.61, size = 1568, normalized size = 3.73

method	result	size
default	\(\text {Expression too large to display}\)	\(1568\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a^2*d^2*x*(c^2*d*x^2+d)^(1/2)+5/16*a^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)+b^2*(5/48*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)
/c*arcsinh(c*x)^3*d^2+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7+32*(c^2*x^2+1)^(1/2)*x^6*c^6+64*x^5*c^5+48*(c^2
*x^2+1)^(1/2)*x^4*c^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))*(18*arcsinh(c*x)^2-6*ar
csinh(c*x)+1)*d^2/c/(c^2*x^2+1)+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*(c^2*x^2+1)^(1/2)*x^4*c^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)^2-4*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+15
/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))*(2*arcsinh(c*x)^2-2
*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+15/256*(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))*(2*arcsinh(c*x)^2+2*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^
5-8*(c^2*x^2+1)^(1/2)*x^4*c^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)^
2+4*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7-32*(c^2*x^2+1)^(1/2)*x^6*c^6+64
*x^5*c^5-48*(c^2*x^2+1)^(1/2)*x^4*c^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))*(18*arc
sinh(c*x)^2+6*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1))+2*a*b*(5/32*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsinh
(c*x)^2*d^2+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7+32*(c^2*x^2+1)^(1/2)*x^6*c^6+64*x^5*c^5+48*(c^2*x^2+1)^(1
/2)*x^4*c^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))*(-1+6*arcsinh(c*x))*d^2/c/(c^2*x^
2+1)+3/512*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*(c^2*x^2+1)^(1/2)*x^4*c^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))*(-1+4*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+15/256*(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))*(-1+2*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+15/256*(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))*(1+2*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+3/512
*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*(c^2*x^2+1)^(1/2)*x^4*c^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))*(1+4*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*x^7*c^7-32*(c^2*x^2+1)^(
1/2)*x^6*c^6+64*x^5*c^5-48*(c^2*x^2+1)^(1/2)*x^4*c^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))*(1+6*arcsinh(c*x))*d^2/c/(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((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,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((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
inh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a*b*c^2*d^2*x^2 + a*b*d^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 {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const 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 (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\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((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2),x)

[Out]

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

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