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

[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)

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Rubi [A]  time = 0.394956, antiderivative size = 225, normalized size of antiderivative = 1., 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 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]

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 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 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 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])

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*}

Mathematica [A]  time = 0.58508, size = 178, normalized size = 0.79 \[ \frac{\sqrt{c d x+d} \sqrt{e-c e x} \left (9 a^2 \left (c^2 x^2-1\right )^2+6 a b c x \sqrt{1-c^2 x^2} \left (c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt{1-c^2 x^2} \left (c^2 x^2-3\right )\right )-2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )+9 b^2 \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)^2\right )}{27 c^2 \left (c^2 x^2-1\right )} \]

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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Maple [F]  time = 0.361, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{cdx+d}\sqrt{-cex+e} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.44801, size = 435, normalized size = 1.93 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.59997, size = 610, normalized size = 2.71 \begin{align*} -\frac{\frac{9 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e} a^{2}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )}{\left | d \right |}}{c d^{3}} + \frac{6 \,{\left (3 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )} \arcsin \left (c x\right ) - \frac{{\left ({\left (c d x + d\right )}^{3} - 3 \,{\left (c d x + d\right )}^{2} d\right )} e^{\left (-\frac{11}{2}\right )}}{d^{\frac{9}{2}}{\left | d \right |}}\right )} a b{\left | d \right |}}{c d^{3}} + \frac{{\left (9 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )} \arcsin \left (c x\right )^{2} + \frac{\sqrt{d}{\left (\frac{6 \, \pi e^{\left (-6\right )}}{d^{2}} - \frac{{\left (6 \,{\left (c^{2} x^{2} - 1\right )} c d^{2} x \arcsin \left (-c x\right ) + 24 \, c d^{2} x \arcsin \left (-c x\right ) - 9 \, \sqrt{-c^{2} x^{2} + 1} c d^{2} x + 18 \,{\left (c^{2} x^{2} - 1\right )} d^{2} \arcsin \left (-c x\right ) + 2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} d^{2} + 9 \, d^{2} \arcsin \left (-c x\right ) - 24 \, \sqrt{-c^{2} x^{2} + 1} d^{2} - 9 \,{\left (4 \, c d x \arcsin \left (-c x\right ) - \sqrt{-c^{2} x^{2} + 1} c d x + 2 \,{\left (c^{2} x^{2} - 1\right )} d \arcsin \left (-c x\right ) + d \arcsin \left (-c x\right ) - 4 \, \sqrt{-c^{2} x^{2} + 1} d\right )} d\right )} e^{\left (-6\right )}}{d^{4}}\right )} e^{\frac{1}{2}}}{{\left | d \right |}}\right )} b^{2}{\left | d \right |}}{c d^{3}}}{4320 \, c d} \end{align*}

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]

-1/4320*(9*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*a^2*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5)*abs(
d)/(c*d^3) + 6*(3*(c*d*x + d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5)*a
rcsin(c*x) - ((c*d*x + d)^3 - 3*(c*d*x + d)^2*d)*e^(-11/2)/(d^(9/2)*abs(d)))*a*b*abs(d)/(c*d^3) + (9*(c*d*x +
d)^(3/2)*sqrt(-(c*d*x + d)*d*e + 2*d^2*e)*((c*d*x + d)*e^(-6)/d^6 - 2*e^(-6)/d^5)*arcsin(c*x)^2 + sqrt(d)*(6*p
i*e^(-6)/d^2 - (6*(c^2*x^2 - 1)*c*d^2*x*arcsin(-c*x) + 24*c*d^2*x*arcsin(-c*x) - 9*sqrt(-c^2*x^2 + 1)*c*d^2*x
+ 18*(c^2*x^2 - 1)*d^2*arcsin(-c*x) + 2*(-c^2*x^2 + 1)^(3/2)*d^2 + 9*d^2*arcsin(-c*x) - 24*sqrt(-c^2*x^2 + 1)*
d^2 - 9*(4*c*d*x*arcsin(-c*x) - sqrt(-c^2*x^2 + 1)*c*d*x + 2*(c^2*x^2 - 1)*d*arcsin(-c*x) + d*arcsin(-c*x) - 4
*sqrt(-c^2*x^2 + 1)*d)*d)*e^(-6)/d^4)*e^(1/2)/abs(d))*b^2*abs(d)/(c*d^3))/(c*d)