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

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

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4739, 4677, 4645, 444, 43} \[ -\frac {2 b c x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {2 b x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {2 b^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c^2}+\frac {4 b^2 \sqrt {c d x+d} \sqrt {e-c e x}}{9 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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]

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.58, size = 178, normalized size = 0.79 \[ \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (9 a^2 \left (c^2 x^2-1\right )^2+6 a b c x \sqrt {1-c^2 x^2} \left (c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt {1-c^2 x^2} \left (c^2 x^2-3\right )\right )+9 b^2 \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)^2-2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )\right )}{27 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 + c^2*x^2) + 9*a^2*(-1 + c^2*x^2)^2 - 2*b^2*
(7 - 8*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 + c^2*x^2) + 3*a*(-1 + c^2*x^2)^2)*ArcSin[c*x] +
9*b^2*(-1 + c^2*x^2)^2*ArcSin[c*x]^2))/(27*c^2*(-1 + c^2*x^2))

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fricas [A]  time = 0.44, size = 197, normalized size = 0.88 \[ \frac {{\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} - 2 \, {\left (9 \, a^{2} - 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + 9 \, a^{2} - 14 \, b^{2} + 18 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right ) + 6 \, {\left (a b c^{3} x^{3} - 3 \, a b c x + {\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{27 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

1/27*((9*a^2 - 2*b^2)*c^4*x^4 - 2*(9*a^2 - 8*b^2)*c^2*x^2 + 9*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arcsin(c*x)^
2 + 9*a^2 - 14*b^2 + 18*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arcsin(c*x) + 6*(a*b*c^3*x^3 - 3*a*b*c*x + (b^2*c^
3*x^3 - 3*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^4*x^2 - c^2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]S
implification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep
near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_
nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Warning, integratio
n of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Sim
plification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep ne
ar 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_no
step near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Warning, integration
of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warni
ng, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs
(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.56, size = 0, normalized size = 0.00 \[ \int x \sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 233, normalized size = 1.04 \[ -\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} e^{\frac {3}{2}} x^{2} - \frac {7 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} e^{\frac {3}{2}}}{c^{2}}}{d e} + \frac {3 \, {\left (c^{2} d^{\frac {3}{2}} e^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} e^{\frac {3}{2}} x\right )} \arcsin \left (c x\right )}{c d e}\right )} - \frac {{\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right )^{2}}{3 \, c^{2} d e} - \frac {2 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} a b \arcsin \left (c x\right )}{3 \, c^{2} d e} - \frac {2 \, {\left (c^{2} d^{\frac {3}{2}} e^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} e^{\frac {3}{2}} x\right )} a b}{9 \, c d e} - \frac {{\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-2/27*b^2*((sqrt(-c^2*x^2 + 1)*d^(3/2)*e^(3/2)*x^2 - 7*sqrt(-c^2*x^2 + 1)*d^(3/2)*e^(3/2)/c^2)/(d*e) + 3*(c^2*
d^(3/2)*e^(3/2)*x^3 - 3*d^(3/2)*e^(3/2)*x)*arcsin(c*x)/(c*d*e)) - 1/3*(-c^2*d*e*x^2 + d*e)^(3/2)*b^2*arcsin(c*
x)^2/(c^2*d*e) - 2/3*(-c^2*d*e*x^2 + d*e)^(3/2)*a*b*arcsin(c*x)/(c^2*d*e) - 2/9*(c^2*d^(3/2)*e^(3/2)*x^3 - 3*d
^(3/2)*e^(3/2)*x)*a*b/(c*d*e) - 1/3*(-c^2*d*e*x^2 + d*e)^(3/2)*a^2/(c^2*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2),x)

[Out]

int(x*(a + b*asin(c*x))^2*(d + c*d*x)^(1/2)*(e - c*e*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)**(1/2)*(-c*e*x+e)**(1/2)*(a+b*asin(c*x))**2,x)

[Out]

Integral(x*sqrt(d*(c*x + 1))*sqrt(-e*(c*x - 1))*(a + b*asin(c*x))**2, x)

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