3.506 \(\int \sqrt{d+c d x} \sqrt{f-c f x} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=134 \[ \frac{\sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(4*Sqrt[1 - c^2*x^2]) + (x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*A
rcSin[c*x]))/2 + (Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.16526, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4673, 4647, 4641, 30} \[ \frac{\sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{c d x+d} \sqrt{f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(4*Sqrt[1 - c^2*x^2]) + (x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*A
rcSin[c*x]))/2 + (Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])

Rule 4673

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> D
ist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(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, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q]
&& GeQ[p - q, 0]

Rule 4647

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.933247, size = 158, normalized size = 1.18 \[ \frac{1}{8} \left (-\frac{4 a \sqrt{d} \sqrt{f} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )}{c}+4 a x \sqrt{c d x+d} \sqrt{f-c f x}+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \left (2 \sin ^{-1}(c x) \left (\sin ^{-1}(c x)+\sin \left (2 \sin ^{-1}(c x)\right )\right )+\cos \left (2 \sin ^{-1}(c x)\right )\right )}{c \sqrt{1-c^2 x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x] - (4*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sq
rt[d]*Sqrt[f]*(-1 + c^2*x^2))])/c + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*(Ar
cSin[c*x] + Sin[2*ArcSin[c*x]])))/(c*Sqrt[1 - c^2*x^2]))/8

________________________________________________________________________________________

Maple [F]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int \sqrt{cdx+d} \left ( a+b\arcsin \left ( cx \right ) \right ) \sqrt{-cfx+f}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*arcsin(c*x) + a), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \left (c x + 1\right )} \sqrt{- f \left (c x - 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*arcsin(c*x) + a), x)