3.6.87 \(\int \frac {x (a+b \text {ArcSin}(c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx\) [587]

Optimal. Leaf size=177 \[ \frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}} \]

[Out]

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

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Rubi [A]
time = 0.26, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4823, 4767, 4715, 267} \begin {gather*} -\frac {\left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*(1 - c^2*x^2))/(c^2*Sqrt[d + c*d*x]*S
qrt[e - c*e*x]) + (2*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - ((1 - c^2*x^2)
*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4715

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 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 4823

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)^Fr
acPart[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 \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 150, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (2 a b c x \sqrt {1-c^2 x^2}+a^2 \left (-1+c^2 x^2\right )-2 b^2 \left (-1+c^2 x^2\right )+2 b \left (b c x \sqrt {1-c^2 x^2}+a \left (-1+c^2 x^2\right )\right ) \text {ArcSin}(c x)+b^2 \left (-1+c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{c^2 d e (-1+c x) (1+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c e x +e}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 157, normalized size = 0.89 \begin {gather*} 2 \, {\left (\frac {x \arcsin \left (c x\right ) e^{\left (-\frac {1}{2}\right )}}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1} e^{\left (-\frac {1}{2}\right )}}{c^{2} \sqrt {d}}\right )} b^{2} + \frac {2 \, a b x e^{\left (-\frac {1}{2}\right )}}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} e + d e} b^{2} \arcsin \left (c x\right )^{2} e^{\left (-1\right )}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} e + d e} a b \arcsin \left (c x\right ) e^{\left (-1\right )}}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} e + d e} a^{2} e^{\left (-1\right )}}{c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x*arcsin(c*x)*e^(-1/2)/(c*sqrt(d)) + sqrt(-c^2*x^2 + 1)*e^(-1/2)/(c^2*sqrt(d)))*b^2 + 2*a*b*x*e^(-1/2)/(c*s
qrt(d)) - sqrt(-c^2*d*x^2*e + d*e)*b^2*arcsin(c*x)^2*e^(-1)/(c^2*d) - 2*sqrt(-c^2*d*x^2*e + d*e)*a*b*arcsin(c*
x)*e^(-1)/(c^2*d) - sqrt(-c^2*d*x^2*e + d*e)*a^2*e^(-1)/(c^2*d)

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Fricas [A]
time = 2.31, size = 153, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {c d x + d} {\left (2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {-{\left (c x - 1\right )} e} + {\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-{\left (c x - 1\right )} e}\right )} e^{\left (-1\right )}}{c^{4} d x^{2} - c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}} \,d x \end {gather*}

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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