3.178 \(\int (d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=298 \[ \frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c}+\frac{32 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 c}+\frac{16}{35} d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}-\frac{4322 b^2 d^3 x}{3675} \]

[Out]

(-4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 - (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 +
(32*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(35*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/
(105*c) + (12*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*Arc
Sin[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSin[c*x])^2)/35 + (8*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (
6*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/35 + (d^3*x*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/7

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Rubi [A]  time = 0.371581, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4649, 4619, 4677, 8, 194} \[ \frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c}+\frac{32 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 c}+\frac{16}{35} d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}-\frac{4322 b^2 d^3 x}{3675} \]

Antiderivative was successfully verified.

[In]

Int[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

(-4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 - (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 +
(32*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(35*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/
(105*c) + (12*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*Arc
Sin[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSin[c*x])^2)/35 + (8*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (
6*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/35 + (d^3*x*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/7

Rule 4649

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4619

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

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 194

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]

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} (6 d) \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{7} \left (2 b c d^3\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{35} \left (24 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{49} \left (2 b^2 d^3\right ) \int \left (1-c^2 x^2\right )^3 \, dx-\frac{1}{35} \left (12 b c d^3\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{35} \left (16 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{49} \left (2 b^2 d^3\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx-\frac{1}{175} \left (12 b^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac{1}{35} \left (16 b c d^3\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{2}{49} b^2 d^3 x+\frac{2}{49} b^2 c^2 d^3 x^3-\frac{6}{245} b^2 c^4 d^3 x^5+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c}+\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{175} \left (12 b^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx-\frac{1}{105} \left (16 b^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac{1}{35} \left (32 b c d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{962 b^2 d^3 x}{3675}+\frac{1514 b^2 c^2 d^3 x^3}{11025}-\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{32 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c}+\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{35} \left (32 b^2 d^3\right ) \int 1 \, dx\\ &=-\frac{4322 b^2 d^3 x}{3675}+\frac{1514 b^2 c^2 d^3 x^3}{11025}-\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{32 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c}+\frac{12 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c}+\frac{16}{35} d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{35} d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{6}{35} d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} d^3 x \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.43053, size = 241, normalized size = 0.81 \[ -\frac{d^3 \left (11025 a^2 c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+210 a b \sqrt{1-c^2 x^2} \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right )+210 b \sin ^{-1}(c x) \left (105 a c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+b \sqrt{1-c^2 x^2} \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right )\right )+2 b^2 c x \left (-1125 c^6 x^6+7371 c^4 x^4-26495 c^2 x^2+226905\right )+11025 b^2 c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right ) \sin ^{-1}(c x)^2\right )}{385875 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

-(d^3*(2*b^2*c*x*(226905 - 26495*c^2*x^2 + 7371*c^4*x^4 - 1125*c^6*x^6) + 11025*a^2*c*x*(-35 + 35*c^2*x^2 - 21
*c^4*x^4 + 5*c^6*x^6) + 210*a*b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 210*b*(10
5*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 +
75*c^6*x^6))*ArcSin[c*x] + 11025*b^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]^2))/(385875*
c)

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Maple [A]  time = 0.042, size = 384, normalized size = 1.3 \begin{align*}{\frac{1}{c} \left ( -{d}^{3}{a}^{2} \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{3\,{c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}-cx \right ) -{d}^{3}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 5\,{c}^{6}{x}^{6}-21\,{c}^{4}{x}^{4}+35\,{c}^{2}{x}^{2}-35 \right ) cx}{35}}+{\frac{32\,cx}{35}}-{\frac{32\,\arcsin \left ( cx \right ) }{35}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{49}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 10\,{c}^{6}{x}^{6}-42\,{c}^{4}{x}^{4}+70\,{c}^{2}{x}^{2}-70 \right ) cx}{1715}}-{\frac{12\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{175}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{ \left ( 12\,{c}^{4}{x}^{4}-40\,{c}^{2}{x}^{2}+60 \right ) cx}{875}}+{\frac{16\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{105}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 16\,{c}^{2}{x}^{2}-48 \right ) cx}{315}} \right ) -2\,{d}^{3}ab \left ( 1/7\,\arcsin \left ( cx \right ){c}^{7}{x}^{7}-3/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}+{c}^{3}{x}^{3}\arcsin \left ( cx \right ) -cx\arcsin \left ( cx \right ) +1/49\,{c}^{6}{x}^{6}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{117\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}}{1225}}+{\frac{757\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{3675}}-{\frac{2161\,\sqrt{-{c}^{2}{x}^{2}+1}}{3675}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x)

[Out]

1/c*(-d^3*a^2*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b^2*(1/35*arcsin(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x
^2-35)*c*x+32/35*c*x-32/35*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/49*arcsin(c*x)*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/
1715*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-12/175*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)+4/875*(3*c^4
*x^4-10*c^2*x^2+15)*c*x+16/105*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/315*(c^2*x^2-3)*c*x)-2*d^3*a*b*(1
/7*arcsin(c*x)*c^7*x^7-3/5*arcsin(c*x)*c^5*x^5+c^3*x^3*arcsin(c*x)-c*x*arcsin(c*x)+1/49*c^6*x^6*(-c^2*x^2+1)^(
1/2)-117/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)+757/3675*c^2*x^2*(-c^2*x^2+1)^(1/2)-2161/3675*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.70148, size = 984, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-1/7*b^2*c^6*d^3*x^7*arcsin(c*x)^2 - 1/7*a^2*c^6*d^3*x^7 + 3/5*b^2*c^4*d^3*x^5*arcsin(c*x)^2 + 3/5*a^2*c^4*d^3
*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^6*d^3 - 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^6*d^3 - b^2*c^2*d^3*x^3*arcsin(c*x)^2 + 2/25*(15*x^5*arc
sin(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*c^4
*d^3 + 2/375*(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*ar
csin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^4*d^3 - a^2*c^2*d^3*x^3 - 2/3*(3*x^3*arcsin(c*x) + c*(
sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*c^2*d^3 - 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2
*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d^3 + b^2*d^3*x*arcsin(c*x)^2 - 2*b^2*d^3*
(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^3*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^3/c

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Fricas [A]  time = 1.93143, size = 759, normalized size = 2.55 \begin{align*} -\frac{1125 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} x^{7} - 189 \,{\left (1225 \, a^{2} - 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \,{\left (11025 \, a^{2} - 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} - 105 \,{\left (3675 \, a^{2} - 4322 \, b^{2}\right )} c d^{3} x + 11025 \,{\left (5 \, b^{2} c^{7} d^{3} x^{7} - 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} - 35 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )^{2} + 22050 \,{\left (5 \, a b c^{7} d^{3} x^{7} - 21 \, a b c^{5} d^{3} x^{5} + 35 \, a b c^{3} d^{3} x^{3} - 35 \, a b c d^{3} x\right )} \arcsin \left (c x\right ) + 210 \,{\left (75 \, a b c^{6} d^{3} x^{6} - 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} - 2161 \, a b d^{3} +{\left (75 \, b^{2} c^{6} d^{3} x^{6} - 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} - 2161 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{385875 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

-1/385875*(1125*(49*a^2 - 2*b^2)*c^7*d^3*x^7 - 189*(1225*a^2 - 78*b^2)*c^5*d^3*x^5 + 35*(11025*a^2 - 1514*b^2)
*c^3*d^3*x^3 - 105*(3675*a^2 - 4322*b^2)*c*d^3*x + 11025*(5*b^2*c^7*d^3*x^7 - 21*b^2*c^5*d^3*x^5 + 35*b^2*c^3*
d^3*x^3 - 35*b^2*c*d^3*x)*arcsin(c*x)^2 + 22050*(5*a*b*c^7*d^3*x^7 - 21*a*b*c^5*d^3*x^5 + 35*a*b*c^3*d^3*x^3 -
35*a*b*c*d^3*x)*arcsin(c*x) + 210*(75*a*b*c^6*d^3*x^6 - 351*a*b*c^4*d^3*x^4 + 757*a*b*c^2*d^3*x^2 - 2161*a*b*
d^3 + (75*b^2*c^6*d^3*x^6 - 351*b^2*c^4*d^3*x^4 + 757*b^2*c^2*d^3*x^2 - 2161*b^2*d^3)*arcsin(c*x))*sqrt(-c^2*x
^2 + 1))/c

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Sympy [A]  time = 21.1895, size = 524, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{a^{2} c^{6} d^{3} x^{7}}{7} + \frac{3 a^{2} c^{4} d^{3} x^{5}}{5} - a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x - \frac{2 a b c^{6} d^{3} x^{7} \operatorname{asin}{\left (c x \right )}}{7} - \frac{2 a b c^{5} d^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49} + \frac{6 a b c^{4} d^{3} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{234 a b c^{3} d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{1225} - 2 a b c^{2} d^{3} x^{3} \operatorname{asin}{\left (c x \right )} - \frac{1514 a b c d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname{asin}{\left (c x \right )} + \frac{4322 a b d^{3} \sqrt{- c^{2} x^{2} + 1}}{3675 c} - \frac{b^{2} c^{6} d^{3} x^{7} \operatorname{asin}^{2}{\left (c x \right )}}{7} + \frac{2 b^{2} c^{6} d^{3} x^{7}}{343} - \frac{2 b^{2} c^{5} d^{3} x^{6} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{49} + \frac{3 b^{2} c^{4} d^{3} x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} - \frac{234 b^{2} c^{4} d^{3} x^{5}}{6125} + \frac{234 b^{2} c^{3} d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{1225} - b^{2} c^{2} d^{3} x^{3} \operatorname{asin}^{2}{\left (c x \right )} + \frac{1514 b^{2} c^{2} d^{3} x^{3}}{11025} - \frac{1514 b^{2} c d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname{asin}^{2}{\left (c x \right )} - \frac{4322 b^{2} d^{3} x}{3675} + \frac{4322 b^{2} d^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3675 c} & \text{for}\: c \neq 0 \\a^{2} d^{3} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((-a**2*c**6*d**3*x**7/7 + 3*a**2*c**4*d**3*x**5/5 - a**2*c**2*d**3*x**3 + a**2*d**3*x - 2*a*b*c**6*d
**3*x**7*asin(c*x)/7 - 2*a*b*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)/49 + 6*a*b*c**4*d**3*x**5*asin(c*x)/5 + 234*a
*b*c**3*d**3*x**4*sqrt(-c**2*x**2 + 1)/1225 - 2*a*b*c**2*d**3*x**3*asin(c*x) - 1514*a*b*c*d**3*x**2*sqrt(-c**2
*x**2 + 1)/3675 + 2*a*b*d**3*x*asin(c*x) + 4322*a*b*d**3*sqrt(-c**2*x**2 + 1)/(3675*c) - b**2*c**6*d**3*x**7*a
sin(c*x)**2/7 + 2*b**2*c**6*d**3*x**7/343 - 2*b**2*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/49 + 3*b**2*c
**4*d**3*x**5*asin(c*x)**2/5 - 234*b**2*c**4*d**3*x**5/6125 + 234*b**2*c**3*d**3*x**4*sqrt(-c**2*x**2 + 1)*asi
n(c*x)/1225 - b**2*c**2*d**3*x**3*asin(c*x)**2 + 1514*b**2*c**2*d**3*x**3/11025 - 1514*b**2*c*d**3*x**2*sqrt(-
c**2*x**2 + 1)*asin(c*x)/3675 + b**2*d**3*x*asin(c*x)**2 - 4322*b**2*d**3*x/3675 + 4322*b**2*d**3*sqrt(-c**2*x
**2 + 1)*asin(c*x)/(3675*c), Ne(c, 0)), (a**2*d**3*x, True))

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Giac [B]  time = 1.48575, size = 713, normalized size = 2.39 \begin{align*} -\frac{1}{7} \, a^{2} c^{6} d^{3} x^{7} + \frac{3}{5} \, a^{2} c^{4} d^{3} x^{5} - \frac{1}{7} \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x \arcsin \left (c x\right )^{2} - a^{2} c^{2} d^{3} x^{3} - \frac{2}{7} \,{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{3} x \arcsin \left (c x\right ) + \frac{6}{35} \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x \arcsin \left (c x\right )^{2} + \frac{2}{343} \,{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x + \frac{12}{35} \,{\left (c^{2} x^{2} - 1\right )}^{2} a b d^{3} x \arcsin \left (c x\right ) - \frac{8}{35} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x \arcsin \left (c x\right )^{2} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{49 \, c} - \frac{888}{42875} \,{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x - \frac{16}{35} \,{\left (c^{2} x^{2} - 1\right )} a b d^{3} x \arcsin \left (c x\right ) + \frac{16}{35} \, b^{2} d^{3} x \arcsin \left (c x\right )^{2} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} a b d^{3}}{49 \, c} + \frac{12 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{175 \, c} + \frac{30256}{385875} \,{\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x + \frac{32}{35} \, a b d^{3} x \arcsin \left (c x\right ) + \frac{12 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d^{3}}{175 \, c} + \frac{16 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d^{3} \arcsin \left (c x\right )}{105 \, c} + a^{2} d^{3} x - \frac{413312}{385875} \, b^{2} d^{3} x + \frac{16 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d^{3}}{105 \, c} + \frac{32 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{35 \, c} + \frac{32 \, \sqrt{-c^{2} x^{2} + 1} a b d^{3}}{35 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

-1/7*a^2*c^6*d^3*x^7 + 3/5*a^2*c^4*d^3*x^5 - 1/7*(c^2*x^2 - 1)^3*b^2*d^3*x*arcsin(c*x)^2 - a^2*c^2*d^3*x^3 - 2
/7*(c^2*x^2 - 1)^3*a*b*d^3*x*arcsin(c*x) + 6/35*(c^2*x^2 - 1)^2*b^2*d^3*x*arcsin(c*x)^2 + 2/343*(c^2*x^2 - 1)^
3*b^2*d^3*x + 12/35*(c^2*x^2 - 1)^2*a*b*d^3*x*arcsin(c*x) - 8/35*(c^2*x^2 - 1)*b^2*d^3*x*arcsin(c*x)^2 - 2/49*
(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c - 888/42875*(c^2*x^2 - 1)^2*b^2*d^3*x - 16/35*(c^2*x^
2 - 1)*a*b*d^3*x*arcsin(c*x) + 16/35*b^2*d^3*x*arcsin(c*x)^2 - 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3
/c + 12/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c + 30256/385875*(c^2*x^2 - 1)*b^2*d^3*x +
32/35*a*b*d^3*x*arcsin(c*x) + 12/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c + 16/105*(-c^2*x^2 + 1)^(3/2
)*b^2*d^3*arcsin(c*x)/c + a^2*d^3*x - 413312/385875*b^2*d^3*x + 16/105*(-c^2*x^2 + 1)^(3/2)*a*b*d^3/c + 32/35*
sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c + 32/35*sqrt(-c^2*x^2 + 1)*a*b*d^3/c