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

Optimal. Leaf size=242 \[ \frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b d e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac {d e \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2}+\frac {2 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {4 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}-\frac {4 b^2 e^2 x}{9 c^2}-2 b^2 d^2 x-\frac {1}{2} b^2 d e x^2-\frac {2}{27} b^2 e^2 x^3 \]

[Out]

-2*b^2*d^2*x-4/9*b^2*e^2*x/c^2-1/2*b^2*d*e*x^2-2/27*b^2*e^2*x^3-1/3*d^3*(a+b*arcsin(c*x))^2/e-1/2*d*e*(a+b*arc
sin(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*arcsin(c*x))^2/e+2*b*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+4/9*b*e^2*(
a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+b*d*e*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c+2/9*b*e^2*x^2*(a+b*arcs
in(c*x))*(-c^2*x^2+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4743, 4763, 4641, 4677, 8, 4707, 30} \[ \frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b d e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac {d e \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2}+\frac {4 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {2 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}-\frac {4 b^2 e^2 x}{9 c^2}-2 b^2 d^2 x-\frac {1}{2} b^2 d e x^2-\frac {2}{27} b^2 e^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*d^2*x - (4*b^2*e^2*x)/(9*c^2) - (b^2*d*e*x^2)/2 - (2*b^2*e^2*x^3)/27 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/c + (4*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (b*d*e*x*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/c + (2*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) - (d^3*(a + b*ArcSin[c*x])^2)/(3
*e) - (d*e*(a + b*ArcSin[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e)

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 4641

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

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]

Rule 4743

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

Rule 4763

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.41, size = 249, normalized size = 1.03 \[ \frac {18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+6 a b \sqrt {1-c^2 x^2} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )+6 b \sin ^{-1}(c x) \left (6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-9 a c d e+b \sqrt {1-c^2 x^2} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )\right )-b^2 c x \left (c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+24 e^2\right )+9 b^2 c \sin ^{-1}(c x)^2 \left (6 c^2 d^2 x+3 d e \left (2 c^2 x^2-1\right )+2 c^2 e^2 x^3\right )}{54 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2
)) - b^2*c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) + 6*b*(-9*a*c*d*e + 6*a*c^3*x*(3*d^2 + 3*d*e*x +
e^2*x^2) + b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcSin[c*x] + 9*b^2*c*(6*c^2*d^2*
x + 2*c^2*e^2*x^3 + 3*d*e*(-1 + 2*c^2*x^2))*ArcSin[c*x]^2)/(54*c^3)

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 290, normalized size = 1.20 \[ \frac {2 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \, {\left (2 \, a^{2} - b^{2}\right )} c^{3} d e x^{2} + 9 \, {\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x - 3 \, b^{2} c d e\right )} \arcsin \left (c x\right )^{2} + 6 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 18 \, {\left (2 \, a b c^{3} e^{2} x^{3} + 6 \, a b c^{3} d e x^{2} + 6 \, a b c^{3} d^{2} x - 3 \, a b c d e\right )} \arcsin \left (c x\right ) + 6 \, {\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} + 4 \, a b e^{2} + {\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} + 4 \, b^{2} e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{54 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(2*(9*a^2 - 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 - b^2)*c^3*d*e*x^2 + 9*(2*b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2
+ 6*b^2*c^3*d^2*x - 3*b^2*c*d*e)*arcsin(c*x)^2 + 6*(9*(a^2 - 2*b^2)*c^3*d^2 - 4*b^2*c*e^2)*x + 18*(2*a*b*c^3*e
^2*x^3 + 6*a*b*c^3*d*e*x^2 + 6*a*b*c^3*d^2*x - 3*a*b*c*d*e)*arcsin(c*x) + 6*(2*a*b*c^2*e^2*x^2 + 9*a*b*c^2*d*e
*x + 18*a*b*c^2*d^2 + 4*a*b*e^2 + (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*d*e*x + 18*b^2*c^2*d^2 + 4*b^2*e^2)*arcsin(c*
x))*sqrt(-c^2*x^2 + 1))/c^3

________________________________________________________________________________________

giac [B]  time = 0.36, size = 485, normalized size = 2.00 \[ b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {1}{3} \, a^{2} x^{3} e^{2} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right ) e}{c} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d \arcsin \left (c x\right )^{2} e}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d x e}{c} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d \arcsin \left (c x\right ) e}{c^{2}} + \frac {b^{2} d \arcsin \left (c x\right )^{2} e}{2 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} x e^{2}}{27 \, c^{2}} + \frac {2 \, a b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d e}{c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d e}{2 \, c^{2}} + \frac {a b d \arcsin \left (c x\right ) e}{c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right ) e^{2}}{9 \, c^{3}} - \frac {14 \, b^{2} x e^{2}}{27 \, c^{2}} - \frac {b^{2} d e}{4 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{9 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right ) e^{2}}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d^2*x*arcsin(c*x)^2 + 2*a*b*d^2*x*arcsin(c*x) + 1/3*a^2*x^3*e^2 + sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)*e
/c + a^2*d^2*x - 2*b^2*d^2*x + 1/3*(c^2*x^2 - 1)*b^2*x*arcsin(c*x)^2*e^2/c^2 + (c^2*x^2 - 1)*b^2*d*arcsin(c*x)
^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c + sqrt(-c^2*x^2 + 1)*a*b*d*x*e/c + 2/3*(c^2*x^2 - 1)*a*b
*x*arcsin(c*x)*e^2/c^2 + 1/3*b^2*x*arcsin(c*x)^2*e^2/c^2 + 2*(c^2*x^2 - 1)*a*b*d*arcsin(c*x)*e/c^2 + 1/2*b^2*d
*arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 2/27*(c^2*x^2 - 1)*b^2*x*e^2/c^2 + 2/3*a*b*x*arcsin(c*
x)*e^2/c^2 + (c^2*x^2 - 1)*a^2*d*e/c^2 - 1/2*(c^2*x^2 - 1)*b^2*d*e/c^2 + a*b*d*arcsin(c*x)*e/c^2 - 2/9*(-c^2*x
^2 + 1)^(3/2)*b^2*arcsin(c*x)*e^2/c^3 - 14/27*b^2*x*e^2/c^2 - 1/4*b^2*d*e/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*e
^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^3

________________________________________________________________________________________

maple [A]  time = 0.11, size = 420, normalized size = 1.74 \[ \frac {\frac {\left (c e x +d c \right )^{3} a^{2}}{3 c^{2} e}+\frac {b^{2} \left (c^{2} d^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {d c e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{2}+\frac {e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}+e \arcsin \left (c x \right ) c^{3} x^{2} d +\arcsin \left (c x \right ) c^{3} x \,d^{2}+\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}-\frac {e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+c^{3} d^{3} \arcsin \left (c x \right )}{3 e}\right )}{c^{2}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(1/3*(c*e*x+c*d)^3*a^2/c^2/e+b^2/c^2*(c^2*d^2*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1
/2*d*c*e*(2*arcsin(c*x)^2*c^2*x^2+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/27*e^2*(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)+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^2*(1/3*arcsin(c*
x)*e^2*c^3*x^3+e*arcsin(c*x)*c^3*x^2*d+arcsin(c*x)*c^3*x*d^2+1/3/e*arcsin(c*x)*c^3*d^3-1/3/e*(e^3*(-1/3*c^2*x^
2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+3*d*c*e^2*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-3*d^2*c^2
*e*(-c^2*x^2+1)^(1/2)+c^3*d^3*arcsin(c*x))))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + a^{2} d e x^{2} + {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d e + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e^{2} - 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} + \frac {1}{3} \, {\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + \int \frac {2 \, {\left (b^{2} c e^{2} x^{3} + 3 \, b^{2} c d e x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*e^2*x^3 + b^2*d^2*x*arcsin(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - a
rcsin(c*x)/c^3))*a*b*d*e + 2/9*(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*e^2 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(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 + 1/3*(b^2*e^2*x^3 + 3*b^2*d*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(2/
3*(b^2*c*e^2*x^3 + 3*b^2*c*d*e*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c
^2*x^2 - 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.57, size = 454, normalized size = 1.88 \[ \begin {cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + 2 a b d e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b e^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b d e x \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {2 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {a b d e \operatorname {asin}{\left (c x \right )}}{c^{2}} + \frac {4 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname {asin}^{2}{\left (c x \right )} - \frac {b^{2} d e x^{2}}{2} + \frac {b^{2} e^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e^{2} x^{3}}{27} + \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} d e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {2 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} - \frac {b^{2} d e \operatorname {asin}^{2}{\left (c x \right )}}{2 c^{2}} - \frac {4 b^{2} e^{2} x}{9 c^{2}} + \frac {4 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*b*d**2*x*asin(c*x) + 2*a*b*d*e*x**2*asin(c*x)
+ 2*a*b*e**2*x**3*asin(c*x)/3 + 2*a*b*d**2*sqrt(-c**2*x**2 + 1)/c + a*b*d*e*x*sqrt(-c**2*x**2 + 1)/c + 2*a*b*e
**2*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - a*b*d*e*asin(c*x)/c**2 + 4*a*b*e**2*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2
*d**2*x*asin(c*x)**2 - 2*b**2*d**2*x + b**2*d*e*x**2*asin(c*x)**2 - b**2*d*e*x**2/2 + b**2*e**2*x**3*asin(c*x)
**2/3 - 2*b**2*e**2*x**3/27 + 2*b**2*d**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + b**2*d*e*x*sqrt(-c**2*x**2 + 1)*a
sin(c*x)/c + 2*b**2*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) - b**2*d*e*asin(c*x)**2/(2*c**2) - 4*b**2*e
**2*x/(9*c**2) + 4*b**2*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3), Ne(c, 0)), (a**2*(d**2*x + d*e*x**2 + e*
*2*x**3/3), True))

________________________________________________________________________________________