3.1.10 \(\int (d+e x)^2 (a+b \text {ArcSin}(c x))^2 \, dx\) [10]

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

[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

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Rubi [A]
time = 0.32, 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 = {4827, 4847, 4737, 4767, 8, 4795, 30} \begin {gather*} \frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {b d e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}-\frac {d e (a+b \text {ArcSin}(c x))^2}{2 c^2}+\frac {2 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {4 b e^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}-\frac {d^3 (a+b \text {ArcSin}(c x))^2}{3 e}+\frac {(d+e x)^3 (a+b \text {ArcSin}(c x))^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 \end {gather*}

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 4737

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4827

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 4847

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*}

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Mathematica [A]
time = 0.20, size = 249, normalized size = 1.03 \begin {gather*} \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 (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )-b^2 c x \left (24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )+6 b \left (-9 a c d e+6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \text {ArcSin}(c x)+9 b^2 c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)^2}{54 c^3} \end {gather*}

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)

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Maple [A]
time = 0.14, size = 420, normalized size = 1.74

method	result	size
derivativedivides	\(\frac {\frac {\left (c e x +d c \right )^{3} a^{2}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {c d e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{2}+d^{2} c^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\)	\(420\)
default	\(\frac {\frac {\left (c e x +d c \right )^{3} a^{2}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {c d e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{2}+d^{2} c^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\)	\(420\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [A]
time = 2.85, size = 289, normalized size = 1.19 \begin {gather*} \frac {27 \, {\left (2 \, a^{2} - b^{2}\right )} c^{3} d x^{2} e + 54 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{3} d^{2} x + 9 \, {\left (2 \, b^{2} c^{3} x^{3} e^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, {\left (2 \, b^{2} c^{3} d x^{2} - b^{2} c d\right )} e\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (2 \, a b c^{3} x^{3} e^{2} + 6 \, a b c^{3} d^{2} x + 3 \, {\left (2 \, a b c^{3} d x^{2} - a b c d\right )} e\right )} \arcsin \left (c x\right ) + 2 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} e^{2} + 6 \, {\left (9 \, a b c^{2} d x e + 18 \, a b c^{2} d^{2} + {\left (9 \, b^{2} c^{2} d x e + 18 \, b^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} e^{2}\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c^{2} x^{2} + 2 \, a b\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{54 \, c^{3}} \end {gather*}

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*(27*(2*a^2 - b^2)*c^3*d*x^2*e + 54*(a^2 - 2*b^2)*c^3*d^2*x + 9*(2*b^2*c^3*x^3*e^2 + 6*b^2*c^3*d^2*x + 3*(
2*b^2*c^3*d*x^2 - b^2*c*d)*e)*arcsin(c*x)^2 + 18*(2*a*b*c^3*x^3*e^2 + 6*a*b*c^3*d^2*x + 3*(2*a*b*c^3*d*x^2 - a
*b*c*d)*e)*arcsin(c*x) + 2*((9*a^2 - 2*b^2)*c^3*x^3 - 12*b^2*c*x)*e^2 + 6*(9*a*b*c^2*d*x*e + 18*a*b*c^2*d^2 +
(9*b^2*c^2*d*x*e + 18*b^2*c^2*d^2 + 2*(b^2*c^2*x^2 + 2*b^2)*e^2)*arcsin(c*x) + 2*(a*b*c^2*x^2 + 2*a*b)*e^2)*sq
rt(-c^2*x^2 + 1))/c^3

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Sympy [A]
time = 0.35, size = 454, normalized size = 1.88 \begin {gather*} \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} \end {gather*}

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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (218) = 436\).
time = 0.40, size = 487, normalized size = 2.01 \begin {gather*} \frac {1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d e x \arcsin \left (c x\right )}{c} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d e \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d e x}{c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x}{27 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d e \arcsin \left (c x\right )}{c^{2}} + \frac {2 \, a b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} d e \arcsin \left (c x\right )^{2}}{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 )}^{\frac {3}{2}} b^{2} e^{2} \arcsin \left (c x\right )}{9 \, c^{3}} + \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 {14 \, b^{2} e^{2} x}{27 \, c^{2}} + \frac {a b d e \arcsin \left (c x\right )}{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} e^{2} \arcsin \left (c x\right )}{3 \, c^{3}} - \frac {b^{2} d e}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{3 \, c^{3}} \end {gather*}

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]

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

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

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)

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