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

Optimal. Leaf size=335 \[ \frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5 \]

[Out]

-2*b^2*d^2*x-8/9*b^2*d*e*x/c^2-16/75*b^2*e^2*x/c^4-4/27*b^2*d*e*x^3-8/225*b^2*e^2*x^3/c^2-2/125*b^2*e^2*x^5+d^
2*x*(a+b*arcsin(c*x))^2+2/3*d*e*x^3*(a+b*arcsin(c*x))^2+1/5*e^2*x^5*(a+b*arcsin(c*x))^2+2*b*d^2*(a+b*arcsin(c*
x))*(-c^2*x^2+1)^(1/2)/c+8/9*b*d*e*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+16/75*b*e^2*(a+b*arcsin(c*x))*(-c^
2*x^2+1)^(1/2)/c^5+4/9*b*d*e*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+8/75*b*e^2*x^2*(a+b*arcsin(c*x))*(-c^2
*x^2+1)^(1/2)/c^3+2/25*b*e^2*x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]  time = 0.56, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4667, 4619, 4677, 8, 4627, 4707, 30} \[ \frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4*b^2*d*e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2
) - (2*b^2*e^2*x^5)/125 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (8*b*d*e*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/(9*c^3) + (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*d*e*x^2*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*e^2*x^
4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + d^2*x*(a + b*ArcSin[c*x])^2 + (2*d*e*x^3*(a + b*ArcSin[c*x])
^2)/3 + (e^2*x^5*(a + b*ArcSin[c*x])^2)/5

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 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 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 4667

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

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \sin ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \sin ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} (4 b c d e) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{5} \left (2 b c e^2\right ) \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d^2\right ) \int 1 \, dx-\frac {1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx-\frac {(8 b d e) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{9 c}-\frac {1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx-\frac {\left (8 b e^2\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{25 c}\\ &=-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac {\left (16 b e^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 c^3}-\frac {\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 291, normalized size = 0.87 \[ \frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+30 a b \sqrt {1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )+30 b \sin ^{-1}(c x) \left (15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )\right )+225 b^2 c^5 x \sin ^{-1}(c x)^2 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-2 b^2 c x \left (c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )+60 c^2 e \left (25 d+e x^2\right )+360 e^2\right )}{3375 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + 30*a*b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2)
+ c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)) - 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 25
0*d*e*x^2 + 27*e^2*x^4)) + 30*b*(15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(24*e^2 +
4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSin[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d
*e*x^2 + 3*e^2*x^4)*ArcSin[c*x]^2)/(3375*c^5)

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fricas [A]  time = 0.99, size = 349, normalized size = 1.04 \[ \frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \arcsin \left (c x\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e - 48 \, b^{2} c e^{2}\right )} x + 450 \, {\left (3 \, a b c^{5} e^{2} x^{5} + 10 \, a b c^{5} d e x^{3} + 15 \, a b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) + 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2} + {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(27*(25*a^2 - 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 - 2*b^2)*c^5*d*e - 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^
5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^2*c^5*d^2*x)*arcsin(c*x)^2 + 15*(225*(a^2 - 2*b^2)*c^5*d^2 - 200*b^2*c^3
*d*e - 48*b^2*c*e^2)*x + 450*(3*a*b*c^5*e^2*x^5 + 10*a*b*c^5*d*e*x^3 + 15*a*b*c^5*d^2*x)*arcsin(c*x) + 30*(9*a
*b*c^4*e^2*x^4 + 225*a*b*c^4*d^2 + 100*a*b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e + 6*a*b*c^2*e^2)*x^2 + (9*
b^2*c^4*e^2*x^4 + 225*b^2*c^4*d^2 + 100*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c^4*d*e + 6*b^2*c^2*e^2)*x^2)*arc
sin(c*x))*sqrt(-c^2*x^2 + 1))/c^5

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giac [B]  time = 0.53, size = 678, normalized size = 2.02 \[ \frac {1}{5} \, a^{2} x^{5} e^{2} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {2}{3} \, a^{2} d x^{3} e + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2} e}{3 \, c^{2}} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {2 \, b^{2} d x \arcsin \left (c x\right )^{2} e}{3 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{5 \, c^{4}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x e}{27 \, c^{2}} + \frac {4 \, a b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right ) e}{9 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{5 \, c^{4}} - \frac {28 \, b^{2} d x e}{27 \, c^{2}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d e}{9 \, c^{3}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right ) e}{3 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} x e^{2}}{125 \, c^{4}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right ) e^{2}}{25 \, c^{5}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d e}{3 \, c^{3}} - \frac {76 \, {\left (c^{2} x^{2} - 1\right )} b^{2} x e^{2}}{1125 \, c^{4}} + \frac {2 \, a b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{25 \, c^{5}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right ) e^{2}}{15 \, c^{5}} - \frac {298 \, b^{2} x e^{2}}{1125 \, c^{4}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{15 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right ) e^{2}}{5 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{5 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5*e^2 + b^2*d^2*x*arcsin(c*x)^2 + 2/3*a^2*d*x^3*e + 2*a*b*d^2*x*arcsin(c*x) + 2/3*(c^2*x^2 - 1)*b^2*
d*x*arcsin(c*x)^2*e/c^2 + a^2*d^2*x - 2*b^2*d^2*x + 4/3*(c^2*x^2 - 1)*a*b*d*x*arcsin(c*x)*e/c^2 + 2/3*b^2*d*x*
arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c + 1/5*(c^2*x^2 - 1)^2*b^2*x*arcsin(c*x)^2*e^2
/c^4 - 4/27*(c^2*x^2 - 1)*b^2*d*x*e/c^2 + 4/3*a*b*d*x*arcsin(c*x)*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 4/9
*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin(c*x)*e/c^3 + 2/5*(c^2*x^2 - 1)^2*a*b*x*arcsin(c*x)*e^2/c^4 + 2/5*(c^2*x^2 -
1)*b^2*x*arcsin(c*x)^2*e^2/c^4 - 28/27*b^2*d*x*e/c^2 - 4/9*(-c^2*x^2 + 1)^(3/2)*a*b*d*e/c^3 + 4/3*sqrt(-c^2*x
^2 + 1)*b^2*d*arcsin(c*x)*e/c^3 - 2/125*(c^2*x^2 - 1)^2*b^2*x*e^2/c^4 + 4/5*(c^2*x^2 - 1)*a*b*x*arcsin(c*x)*e^
2/c^4 + 1/5*b^2*x*arcsin(c*x)^2*e^2/c^4 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^5 + 4/
3*sqrt(-c^2*x^2 + 1)*a*b*d*e/c^3 - 76/1125*(c^2*x^2 - 1)*b^2*x*e^2/c^4 + 2/5*a*b*x*arcsin(c*x)*e^2/c^4 + 2/25*
(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b^2*arcsin(c*x)*e^2/c^5 - 298/1125*
b^2*x*e^2/c^4 - 4/15*(-c^2*x^2 + 1)^(3/2)*a*b*e^2/c^5 + 2/5*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^5 + 2/5*s
qrt(-c^2*x^2 + 1)*a*b*e^2/c^5

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maple [B]  time = 0.13, size = 635, normalized size = 1.90 \[ \frac {\frac {a^{2} \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} e d \,x^{3}+d^{2} c^{5} x \right )}{c^{4}}+\frac {b^{2} \left (\frac {e^{2} \left (675 \arcsin \left (c x \right )^{2} c^{5} x^{5}+270 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2250 \arcsin \left (c x \right )^{2} c^{3} x^{3}-54 c^{5} x^{5}-1140 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+3375 c x \arcsin \left (c x \right )^{2}+380 c^{3} x^{3}+4470 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4470 c x \right )}{3375}+\frac {2 c^{2} e d \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {2 e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+d^{2} c^{4} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 c^{2} e d \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \arcsin \left (c x \right ) c^{5} e d \,x^{3}}{3}+\arcsin \left (c x \right ) d^{2} c^{5} x -\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {2 c^{2} e d \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^4*(1/5*e^2*c^5*x^5+2/3*c^5*e*d*x^3+d^2*c^5*x)+b^2/c^4*(1/3375*e^2*(675*arcsin(c*x)^2*c^5*x^5+270*ar
csin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-2250*arcsin(c*x)^2*c^3*x^3-54*c^5*x^5-1140*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^2*x^2+3375*c*x*arcsin(c*x)^2+380*c^3*x^3+4470*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4470*c*x)+2/27*c^2*e*d*(9*arcs
in(c*x)^2*c^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*c*x*arcsin(c*x)^2-2*c^3*x^3-42*arcsin(c*x)*(-c^2
*x^2+1)^(1/2)+42*c*x)+2/27*e^2*(9*arcsin(c*x)^2*c^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-27*c*x*arcsin
(c*x)^2-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+d^2*c^4*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c
^2*x^2+1)^(1/2))+2*c^2*e*d*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+e^2*(c*x*arcsin(c*x)^2-2
*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c^4*(1/5*arcsin(c*x)*e^2*c^5*x^5+2/3*arcsin(c*x)*c^5*e*d*x^3+arc
sin(c*x)*d^2*c^5*x-1/5*e^2*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^
(1/2))-2/3*c^2*e*d*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+d^2*c^4*(-c^2*x^2+1)^(1/2)))

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maxima [A]  time = 0.45, size = 437, normalized size = 1.30 \[ \frac {1}{5} \, b^{2} e^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \arcsin \left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e + \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} - 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*e^2*x^5*arcsin(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arcsin(c*x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*
arcsin(c*x)^2 + 4/9*(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*d*e +
4/27*(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*d*e +
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*e^2 + 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*e^2 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*ar
csin(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.61, size = 595, normalized size = 1.78 \[ \begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {4 a b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 a b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {16 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} e^{2} x^{5}}{125} + \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {4 b^{2} d e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {2 b^{2} e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{3}} - \frac {16 b^{2} e^{2} x}{75 c^{4}} + \frac {16 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2*x*asin(c*x) + 4*a*b*d*e*x**3*asin(c
*x)/3 + 2*a*b*e**2*x**5*asin(c*x)/5 + 2*a*b*d**2*sqrt(-c**2*x**2 + 1)/c + 4*a*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/
(9*c) + 2*a*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 8*a*b*d*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*a*b*e**2*x**
2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 16*a*b*e**2*sqrt(-c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*asin(c*x)**2 - 2*b
**2*d**2*x + 2*b**2*d*e*x**3*asin(c*x)**2/3 - 4*b**2*d*e*x**3/27 + b**2*e**2*x**5*asin(c*x)**2/5 - 2*b**2*e**2
*x**5/125 + 2*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 4*b**2*d*e*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c
) + 2*b**2*e**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(25*c) - 8*b**2*d*e*x/(9*c**2) - 8*b**2*e**2*x**3/(225*c**
2) + 8*b**2*d*e*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3) + 8*b**2*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*
c**3) - 16*b**2*e**2*x/(75*c**4) + 16*b**2*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(75*c**5), Ne(c, 0)), (a**2*(d*
*2*x + 2*d*e*x**3/3 + e**2*x**5/5), True))

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