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]

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

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Rubi [A]  time = 0.37, antiderivative size = 298, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.45, 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]

-1/385875*(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*(105*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))
/c

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fricas [A]  time = 0.54, size = 323, normalized size = 1.08 \[ -\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} x^{7} - 189 \, {\left (1225 \, a^{2} - 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} - 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} - 105 \, {\left (3675 \, a^{2} - 4322 \, b^{2}\right )} c d^{3} x + 11025 \, {\left (5 \, b^{2} c^{7} d^{3} x^{7} - 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} - 35 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )^{2} + 22050 \, {\left (5 \, a b c^{7} d^{3} x^{7} - 21 \, a b c^{5} d^{3} x^{5} + 35 \, a b c^{3} d^{3} x^{3} - 35 \, a b c d^{3} x\right )} \arcsin \left (c x\right ) + 210 \, {\left (75 \, a b c^{6} d^{3} x^{6} - 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} - 2161 \, a b d^{3} + {\left (75 \, b^{2} c^{6} d^{3} x^{6} - 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} - 2161 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c} \]

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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giac [B]  time = 0.61, size = 528, normalized size = 1.77 \[ -\frac {1}{7} \, a^{2} c^{6} d^{3} x^{7} + \frac {3}{5} \, a^{2} c^{4} d^{3} x^{5} - \frac {1}{7} \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x \arcsin \left (c x\right )^{2} - a^{2} c^{2} d^{3} x^{3} - \frac {2}{7} \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{3} x \arcsin \left (c x\right ) + \frac {6}{35} \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x \arcsin \left (c x\right )^{2} + \frac {2}{343} \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x + \frac {12}{35} \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{3} x \arcsin \left (c x\right ) - \frac {8}{35} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x \arcsin \left (c x\right )^{2} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{49 \, c} - \frac {888}{42875} \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x - \frac {16}{35} \, {\left (c^{2} x^{2} - 1\right )} a b d^{3} x \arcsin \left (c x\right ) + \frac {16}{35} \, b^{2} d^{3} x \arcsin \left (c x\right )^{2} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{49 \, c} + \frac {12 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{175 \, c} + \frac {30256}{385875} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x + \frac {32}{35} \, a b d^{3} x \arcsin \left (c x\right ) + \frac {12 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{175 \, c} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} \arcsin \left (c x\right )}{105 \, c} + a^{2} d^{3} x - \frac {413312}{385875} \, b^{2} d^{3} x + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3}}{105 \, c} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{35 \, c} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{35 \, c} \]

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

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maple [A]  time = 0.06, size = 384, normalized size = 1.29 \[ \frac {-d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {32 c x}{35}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}-\frac {12 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c} \]

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 = 0.72, size = 729, normalized size = 2.45 \[ -\frac {1}{7} \, b^{2} c^{6} d^{3} x^{7} \arcsin \left (c x\right )^{2} - \frac {1}{7} \, a^{2} c^{6} d^{3} x^{7} + \frac {3}{5} \, b^{2} c^{4} d^{3} x^{5} \arcsin \left (c x\right )^{2} + \frac {3}{5} \, a^{2} c^{4} d^{3} 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^{6} d^{3} - \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^{6} d^{3} - b^{2} c^{2} d^{3} x^{3} \arcsin \left (c x\right )^{2} + \frac {2}{25} \, {\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 c^{4} d^{3} + \frac {2}{375} \, {\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} c^{4} d^{3} - a^{2} c^{2} d^{3} x^{3} - \frac {2}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{3} - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{3} + b^{2} d^{3} x \arcsin \left (c x\right )^{2} - 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 12.52, size = 524, normalized size = 1.76 \[ \begin {cases} - \frac {a^{2} c^{6} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{4} d^{3} x^{5}}{5} - a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x - \frac {2 a b c^{6} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {2 a b c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {6 a b c^{4} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {234 a b c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} - 2 a b c^{2} d^{3} x^{3} \operatorname {asin}{\left (c x \right )} - \frac {1514 a b c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname {asin}{\left (c x \right )} + \frac {4322 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{3675 c} - \frac {b^{2} c^{6} d^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{6} d^{3} x^{7}}{343} - \frac {2 b^{2} c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{49} + \frac {3 b^{2} c^{4} d^{3} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {234 b^{2} c^{4} d^{3} x^{5}}{6125} + \frac {234 b^{2} c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1225} - b^{2} c^{2} d^{3} x^{3} \operatorname {asin}^{2}{\left (c x \right )} + \frac {1514 b^{2} c^{2} d^{3} x^{3}}{11025} - \frac {1514 b^{2} c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname {asin}^{2}{\left (c x \right )} - \frac {4322 b^{2} d^{3} x}{3675} + \frac {4322 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675 c} & \text {for}\: c \neq 0 \\a^{2} d^{3} x & \text {otherwise} \end {cases} \]

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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