3.156 \(\int x^4 (d-c^2 d x^2) (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=290 \[ \frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}+\frac{32 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^2 d x^7-\frac{152 b^2 d x^3}{11025 c^2}-\frac{304 b^2 d x}{3675 c^4}-\frac{38 b^2 d x^5}{6125} \]

[Out]

(-304*b^2*d*x)/(3675*c^4) - (152*b^2*d*x^3)/(11025*c^2) - (38*b^2*d*x^5)/6125 + (2*b^2*c^2*d*x^7)/343 + (32*b*
d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c^5) + (16*b*d*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c
^3) + (4*b*d*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x
]))/(21*c^5) - (4*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*c^5) + (2*b*d*(1 - c^2*x^2)^(7/2)*(a + b*Ar
cSin[c*x]))/(49*c^5) + (2*d*x^5*(a + b*ArcSin[c*x])^2)/35 + (d*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7

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Rubi [A]  time = 0.461145, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4699, 4627, 4707, 4677, 8, 30, 266, 43, 4689, 12} \[ \frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}+\frac{32 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^2 d x^7-\frac{152 b^2 d x^3}{11025 c^2}-\frac{304 b^2 d x}{3675 c^4}-\frac{38 b^2 d x^5}{6125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-304*b^2*d*x)/(3675*c^4) - (152*b^2*d*x^3)/(11025*c^2) - (38*b^2*d*x^5)/6125 + (2*b^2*c^2*d*x^7)/343 + (32*b*
d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c^5) + (16*b*d*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c
^3) + (4*b*d*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x
]))/(21*c^5) - (4*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*c^5) + (2*b*d*(1 - c^2*x^2)^(7/2)*(a + b*Ar
cSin[c*x]))/(49*c^5) + (2*d*x^5*(a + b*ArcSin[c*x])^2)/35 + (d*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7

Rule 4699

Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n -
1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[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 4707

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

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*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 8

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

Rule 30

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4689

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

Rule 12

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

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} (2 d) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{7} (2 b c d) \int x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{35} (4 b c d) \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{7} \left (2 b^2 c^2 d\right ) \int \frac{-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{175} \left (4 b^2 d\right ) \int x^4 \, dx+\frac{\left (2 b^2 d\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{735 c^4}-\frac{(16 b d) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{175 c}\\ &=-\frac{16 b^2 d x}{735 c^4}-\frac{8 b^2 d x^3}{2205 c^2}-\frac{38 b^2 d x^5}{6125}+\frac{2}{343} b^2 c^2 d x^7+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{(32 b d) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{525 c^3}-\frac{\left (16 b^2 d\right ) \int x^2 \, dx}{525 c^2}\\ &=-\frac{16 b^2 d x}{735 c^4}-\frac{152 b^2 d x^3}{11025 c^2}-\frac{38 b^2 d x^5}{6125}+\frac{2}{343} b^2 c^2 d x^7+\frac{32 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (32 b^2 d\right ) \int 1 \, dx}{525 c^4}\\ &=-\frac{304 b^2 d x}{3675 c^4}-\frac{152 b^2 d x^3}{11025 c^2}-\frac{38 b^2 d x^5}{6125}+\frac{2}{343} b^2 c^2 d x^7+\frac{32 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac{4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac{2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac{2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.265399, size = 203, normalized size = 0.7 \[ -\frac{d \left (11025 a^2 c^5 x^5 \left (5 c^2 x^2-7\right )+210 a b \sqrt{1-c^2 x^2} \left (75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right )+210 b \sin ^{-1}(c x) \left (105 a c^5 x^5 \left (5 c^2 x^2-7\right )+b \sqrt{1-c^2 x^2} \left (75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right )\right )+b^2 \left (-2250 c^7 x^7+2394 c^5 x^5+5320 c^3 x^3+31920 c x\right )+11025 b^2 c^5 x^5 \left (5 c^2 x^2-7\right ) \sin ^{-1}(c x)^2\right )}{385875 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(11025*a^2*c^5*x^5*(-7 + 5*c^2*x^2) + 210*a*b*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x
^6) + b^2*(31920*c*x + 5320*c^3*x^3 + 2394*c^5*x^5 - 2250*c^7*x^7) + 210*b*(105*a*c^5*x^5*(-7 + 5*c^2*x^2) + b
*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6))*ArcSin[c*x] + 11025*b^2*c^5*x^5*(-7 + 5*c^2*
x^2)*ArcSin[c*x]^2))/(385875*c^5)

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Maple [A]  time = 0.111, size = 276, normalized size = 1. \begin{align*}{\frac{1}{{c}^{5}} \left ( -d{a}^{2} \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{{c}^{5}{x}^{5}}{5}} \right ) -d{b}^{2} \left ( -{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{5}{x}^{5}}{5}}-{\frac{2\,\arcsin \left ( cx \right ) \left ( 3\,{c}^{4}{x}^{4}+4\,{c}^{2}{x}^{2}+8 \right ) }{75}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{38\,{c}^{5}{x}^{5}}{6125}}+{\frac{152\,{c}^{3}{x}^{3}}{11025}}+{\frac{304\,cx}{3675}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,\arcsin \left ( cx \right ) \left ( 5\,{c}^{6}{x}^{6}+6\,{c}^{4}{x}^{4}+8\,{c}^{2}{x}^{2}+16 \right ) }{245}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{7}{x}^{7}}{343}} \right ) -2\,dab \left ( 1/7\,\arcsin \left ( cx \right ){c}^{7}{x}^{7}-1/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}+1/49\,{c}^{6}{x}^{6}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{19\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}}{1225}}-{\frac{76\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{3675}}-{\frac{152\,\sqrt{-{c}^{2}{x}^{2}+1}}{3675}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x)

[Out]

1/c^5*(-d*a^2*(1/7*c^7*x^7-1/5*c^5*x^5)-d*b^2*(-1/5*arcsin(c*x)^2*c^5*x^5-2/75*arcsin(c*x)*(3*c^4*x^4+4*c^2*x^
2+8)*(-c^2*x^2+1)^(1/2)+38/6125*c^5*x^5+152/11025*c^3*x^3+304/3675*c*x+1/7*arcsin(c*x)^2*c^7*x^7+2/245*arcsin(
c*x)*(5*c^6*x^6+6*c^4*x^4+8*c^2*x^2+16)*(-c^2*x^2+1)^(1/2)-2/343*c^7*x^7)-2*d*a*b*(1/7*arcsin(c*x)*c^7*x^7-1/5
*arcsin(c*x)*c^5*x^5+1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)-19/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)-76/3675*c^2*x^2*(-c^2*
x^2+1)^(1/2)-152/3675*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.68027, size = 612, normalized size = 2.11 \begin{align*} -\frac{1}{7} \, b^{2} c^{2} d x^{7} \arcsin \left (c x\right )^{2} - \frac{1}{7} \, a^{2} c^{2} d x^{7} + \frac{1}{5} \, b^{2} d x^{5} \arcsin \left (c x\right )^{2} + \frac{1}{5} \, a^{2} d x^{5} - \frac{2}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d - \frac{2}{25725} \,{\left (105 \,{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac{75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d + \frac{2}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d + \frac{2}{1125} \,{\left (15 \,{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/7*b^2*c^2*d*x^7*arcsin(c*x)^2 - 1/7*a^2*c^2*d*x^7 + 1/5*b^2*d*x^5*arcsin(c*x)^2 + 1/5*a^2*d*x^5 - 2/245*(35
*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6
+ 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^2*d - 2/25725*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)
*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcsin(c*x) - (75*c^6*x^7 + 126*c^4*x^5
+ 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^2*d + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c
^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*d + 2/1125*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-
c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arcsin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*d

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Fricas [A]  time = 1.89549, size = 559, normalized size = 1.93 \begin{align*} -\frac{1125 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d x^{7} - 63 \,{\left (1225 \, a^{2} - 38 \, b^{2}\right )} c^{5} d x^{5} + 5320 \, b^{2} c^{3} d x^{3} + 31920 \, b^{2} c d x + 11025 \,{\left (5 \, b^{2} c^{7} d x^{7} - 7 \, b^{2} c^{5} d x^{5}\right )} \arcsin \left (c x\right )^{2} + 22050 \,{\left (5 \, a b c^{7} d x^{7} - 7 \, a b c^{5} d x^{5}\right )} \arcsin \left (c x\right ) + 210 \,{\left (75 \, a b c^{6} d x^{6} - 57 \, a b c^{4} d x^{4} - 76 \, a b c^{2} d x^{2} - 152 \, a b d +{\left (75 \, b^{2} c^{6} d x^{6} - 57 \, b^{2} c^{4} d x^{4} - 76 \, b^{2} c^{2} d x^{2} - 152 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{385875 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

-1/385875*(1125*(49*a^2 - 2*b^2)*c^7*d*x^7 - 63*(1225*a^2 - 38*b^2)*c^5*d*x^5 + 5320*b^2*c^3*d*x^3 + 31920*b^2
*c*d*x + 11025*(5*b^2*c^7*d*x^7 - 7*b^2*c^5*d*x^5)*arcsin(c*x)^2 + 22050*(5*a*b*c^7*d*x^7 - 7*a*b*c^5*d*x^5)*a
rcsin(c*x) + 210*(75*a*b*c^6*d*x^6 - 57*a*b*c^4*d*x^4 - 76*a*b*c^2*d*x^2 - 152*a*b*d + (75*b^2*c^6*d*x^6 - 57*
b^2*c^4*d*x^4 - 76*b^2*c^2*d*x^2 - 152*b^2*d)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 17.71, size = 388, normalized size = 1.34 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} d x^{7}}{7} + \frac{a^{2} d x^{5}}{5} - \frac{2 a b c^{2} d x^{7} \operatorname{asin}{\left (c x \right )}}{7} - \frac{2 a b c d x^{6} \sqrt{- c^{2} x^{2} + 1}}{49} + \frac{2 a b d x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{38 a b d x^{4} \sqrt{- c^{2} x^{2} + 1}}{1225 c} + \frac{152 a b d x^{2} \sqrt{- c^{2} x^{2} + 1}}{3675 c^{3}} + \frac{304 a b d \sqrt{- c^{2} x^{2} + 1}}{3675 c^{5}} - \frac{b^{2} c^{2} d x^{7} \operatorname{asin}^{2}{\left (c x \right )}}{7} + \frac{2 b^{2} c^{2} d x^{7}}{343} - \frac{2 b^{2} c d x^{6} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{49} + \frac{b^{2} d x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} - \frac{38 b^{2} d x^{5}}{6125} + \frac{38 b^{2} d x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{1225 c} - \frac{152 b^{2} d x^{3}}{11025 c^{2}} + \frac{152 b^{2} d x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3675 c^{3}} - \frac{304 b^{2} d x}{3675 c^{4}} + \frac{304 b^{2} d \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3675 c^{5}} & \text{for}\: c \neq 0 \\\frac{a^{2} d x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((-a**2*c**2*d*x**7/7 + a**2*d*x**5/5 - 2*a*b*c**2*d*x**7*asin(c*x)/7 - 2*a*b*c*d*x**6*sqrt(-c**2*x**
2 + 1)/49 + 2*a*b*d*x**5*asin(c*x)/5 + 38*a*b*d*x**4*sqrt(-c**2*x**2 + 1)/(1225*c) + 152*a*b*d*x**2*sqrt(-c**2
*x**2 + 1)/(3675*c**3) + 304*a*b*d*sqrt(-c**2*x**2 + 1)/(3675*c**5) - b**2*c**2*d*x**7*asin(c*x)**2/7 + 2*b**2
*c**2*d*x**7/343 - 2*b**2*c*d*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/49 + b**2*d*x**5*asin(c*x)**2/5 - 38*b**2*d*
x**5/6125 + 38*b**2*d*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(1225*c) - 152*b**2*d*x**3/(11025*c**2) + 152*b**2*d
*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3675*c**3) - 304*b**2*d*x/(3675*c**4) + 304*b**2*d*sqrt(-c**2*x**2 + 1)*
asin(c*x)/(3675*c**5), Ne(c, 0)), (a**2*d*x**5/5, True))

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Giac [A]  time = 1.46853, size = 668, normalized size = 2.3 \begin{align*} -\frac{1}{7} \, a^{2} c^{2} d x^{7} + \frac{1}{5} \, a^{2} d x^{5} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x \arcsin \left (c x\right )^{2}}{7 \, c^{4}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} a b d x \arcsin \left (c x\right )}{7 \, c^{4}} - \frac{8 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x}{343 \, c^{4}} - \frac{16 \,{\left (c^{2} x^{2} - 1\right )}^{2} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{49 \, c^{5}} + \frac{484 \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x}{42875 \, c^{4}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} + \frac{2 \, b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} a b d}{49 \, c^{5}} - \frac{16 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{175 \, c^{5}} - \frac{3358 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d x}{385875 \, c^{4}} + \frac{4 \, a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac{16 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d}{175 \, c^{5}} + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d \arcsin \left (c x\right )}{105 \, c^{5}} - \frac{37384 \, b^{2} d x}{385875 \, c^{4}} + \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d}{105 \, c^{5}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{35 \, c^{5}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} a b d}{35 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

-1/7*a^2*c^2*d*x^7 + 1/5*a^2*d*x^5 - 1/7*(c^2*x^2 - 1)^3*b^2*d*x*arcsin(c*x)^2/c^4 - 2/7*(c^2*x^2 - 1)^3*a*b*d
*x*arcsin(c*x)/c^4 - 8/35*(c^2*x^2 - 1)^2*b^2*d*x*arcsin(c*x)^2/c^4 + 2/343*(c^2*x^2 - 1)^3*b^2*d*x/c^4 - 16/3
5*(c^2*x^2 - 1)^2*a*b*d*x*arcsin(c*x)/c^4 - 1/35*(c^2*x^2 - 1)*b^2*d*x*arcsin(c*x)^2/c^4 - 2/49*(c^2*x^2 - 1)^
3*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)/c^5 + 484/42875*(c^2*x^2 - 1)^2*b^2*d*x/c^4 - 2/35*(c^2*x^2 - 1)*a*b*d*
x*arcsin(c*x)/c^4 + 2/35*b^2*d*x*arcsin(c*x)^2/c^4 - 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d/c^5 - 16/17
5*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)/c^5 - 3358/385875*(c^2*x^2 - 1)*b^2*d*x/c^4 + 4/35*a*b*
d*x*arcsin(c*x)/c^4 - 16/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d/c^5 + 2/105*(-c^2*x^2 + 1)^(3/2)*b^2*d*a
rcsin(c*x)/c^5 - 37384/385875*b^2*d*x/c^4 + 2/105*(-c^2*x^2 + 1)^(3/2)*a*b*d/c^5 + 4/35*sqrt(-c^2*x^2 + 1)*b^2
*d*arcsin(c*x)/c^5 + 4/35*sqrt(-c^2*x^2 + 1)*a*b*d/c^5