3.1.42 \(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \text {ArcSin}(c x)) \, dx\) [42]

Optimal. Leaf size=517 \[ \frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{32 b c \sqrt {1-c^2 x^2}} \]

[Out]

1/36*b*d^2*f*(-c^2*x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+5/2
4*d^2*f*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c
^2*d*x^2+d)^(1/2)-1/7*d^2*g*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/7*b*d^2*g*x*(-c^2*d*x^
2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-25/96*b*c*d^2*f*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/7*b*c*d^2*g*x^3*
(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/96*b*c^3*d^2*f*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+3/35*b*c^
3*d^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/49*b*c^5*d^2*g*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/
2)+5/32*d^2*f*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.27, antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {4861, 4847, 4743, 4741, 4737, 30, 14, 267, 4767, 200} \begin {gather*} \frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{7 c^2}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 -
c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2]
)/(96*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g*x^7*S
qrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5*d^2
*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/16 + (5*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x]))/24 + (d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/6 - (d^2*g*(1 - c^2*x^2)^3*Sqrt[
d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c*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 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 4737

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

Rule 4741

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

Rule 4743

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

Rule 4767

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

Rule 4847

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

Rule 4861

Int[((a_.) + ArcSin[(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*ArcSin[c*x])^n, x], x] /; F
reeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

Rubi steps

\begin {align*} \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-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 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 f \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 (b d^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 {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f \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 (b d^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 {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-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 \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 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{24} d^2 f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^2 f x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 251, normalized size = 0.49 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (11025 a^2 c f+210 a b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+b^2 c x \left (-245 c^2 f x \left (99-39 c^2 x^2+8 c^4 x^4\right )-288 g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right )+210 b \left (105 a c f+b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )\right ) \text {ArcSin}(c x)+11025 b^2 c f \text {ArcSin}(c x)^2\right )}{70560 b c^2 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(11025*a^2*c*f + 210*a*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 2
6*c^2*x^2 + 8*c^4*x^4)) + b^2*c*x*(-245*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) - 288*g*(-35 + 35*c^2*x^2 - 21*c
^4*x^4 + 5*c^6*x^6)) + 210*b*(105*a*c*f + b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*
x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 11025*b^2*c*f*ArcSin[c*x]^2))/(70560*b*c^2*Sqrt[1 - c^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.53, size = 1423, normalized size = 2.75

method	result	size
default	\(\text {Expression too large to display}\)	\(1423\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a*g/c^2/d*(-c^2*d*x^2+d)^(7/2)+1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*f*
d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-5/32*(-
d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*f*d^2+1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^
8*x^8-144*c^6*x^6-64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I
*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g*(I+7*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(
c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I
*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*f*(I+6*arcsin(c*x))*d^2/c/(c^2*x^2-1)-5/128
*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(
c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(arcsin(c*x)-I)*d^2/c^2/(c^2*x^2-1)+15/256*(-d*(c^2*x
^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-I+2*arcsin(c*x))*d^2/c/
(c^2*x^2-1)+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-
5*c^2*x^2+1)*g*(-I+3*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/7840*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+
c^2*x^2-1)*g*(11*I+70*arcsin(c*x))*cos(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3/15680*(-d*(c^2*x^2-1))^(1/2)*(I*x^
2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(9*I+35*arcsin(c*x))*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/4608*(-d*(c^2*
x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*(29*I+96*arcsin(c*x))*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1
)+5/4608*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*(5*I+24*arcsin(c*x))*sin(5*arcsin(c*x))
*d^2/c/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(I+5*arcsin(c*x))*cos(4
*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/320*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(3*I+5*a
rcsin(c*x))*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3/512*(-d*(c^2*x^2-1))^(1/2)*(I*x^2*c^2-c*x*(-c^2*x^2+1)^(1
/2)-I)*f*(11*I+16*arcsin(c*x))*cos(3*arcsin(c*x))*d^2/c/(c^2*x^2-1)+9/512*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+
1)^(1/2)*x*c+c^2*x^2-1)*f*(3*I+8*arcsin(c*x))*sin(3*arcsin(c*x))*d^2/c/(c^2*x^2-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a*g/(c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g*x^5 + b*c^4*d^2*f
*x^4 - 2*b*c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1)), x)

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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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 - 2*a*c^2*d^2*g*x^3 - 2*a*c^2*d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b
*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 - 2*b*c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*arcsin(c*x))*s
qrt(-c^2*d*x^2 + d), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(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 )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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