3.49 \(\int \frac{(f+g x)^3 (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=315 \[ \frac{\left (c^2 f x \left (c^2 f^2+3 g^2\right )+g \left (3 c^2 f^2+g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 \log (c x+1)}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g)^3 \log (1-c x)}{2 c^4 d \sqrt{d-c^2 d x^2}}-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}} \]

[Out]

-((b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2])) + ((g*(3*c^2*f^2 + g^2) + c^2*f*(c^2*f^2 + 3*g^2)*x
)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) + (g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c
^2*d*x^2]) - (3*f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^
3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Log[1 + c*x
])/(2*c^4*d*Sqrt[d - c^2*d*x^2])

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Rubi [A]  time = 0.555683, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {4777, 4775, 637, 4761, 12, 633, 31, 4641, 4677, 8} \[ \frac{\left (c^2 f x \left (c^2 f^2+3 g^2\right )+g \left (3 c^2 f^2+g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 \log (c x+1)}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g)^3 \log (1-c x)}{2 c^4 d \sqrt{d-c^2 d x^2}}-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]

[Out]

-((b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2])) + ((g*(3*c^2*f^2 + g^2) + c^2*f*(c^2*f^2 + 3*g^2)*x
)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) + (g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c
^2*d*x^2]) - (3*f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^
3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Log[1 + c*x
])/(2*c^4*d*Sqrt[d - c^2*d*x^2])

Rule 4777

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

Rule 4775

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 4761

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With
[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 - c^
2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2,
0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 f g^2 \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{1-c^2 x^2}}-\frac{g^3 x \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{1-c^2 x^2}}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \frac{\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 f g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (g^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x}{c^2 \left (1-c^2 x^2\right )} \, dx}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (b g^3 \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x}{1-c^2 x^2} \, dx}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{-c-c^2 x} \, dx}{2 c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{c-c^2 x} \, dx}{2 c^2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (g \left (3 c^2 f^2+g^2\right )+c^2 f \left (c^2 f^2+3 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \log (1-c x)}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.03947, size = 194, normalized size = 0.62 \[ \frac{\sqrt{1-c^2 x^2} \left (2 g^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3 c f g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}+(c f-g)^3 \left (2 b \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-\cot \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )\right )+(c f+g)^3 \left (\tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b \log \left (\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )-2 b c g^3 x\right )}{2 c^4 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]

[Out]

(Sqrt[1 - c^2*x^2]*(-2*b*c*g^3*x + 2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (3*c*f*g^2*(a + b*ArcSin[c*x]
)^2)/b + (c*f - g)^3*(-((a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcSin[c*x])/4]
]) + (c*f + g)^3*(2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2
*c^4*d*Sqrt[d - c^2*d*x^2])

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Maple [C]  time = 0.518, size = 1158, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x)

[Out]

-a*g^3*x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2*a*g^3/d/c^4/(-c^2*d*x^2+d)^(1/2)+3*a*f*g^2*x/c^2/d/(-c^2*d*x^2+d)^(1/2
)-3*a*f*g^2/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+3*a*f^2*g/c^2/d/(-c^2*d*x^2+d)^(1
/2)+a*f^3/d*x/(-c^2*d*x^2+d)^(1/2)-2*b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)*arcsin(c*x)+3*I*b*(-c^2*
x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(c^2*x^2-1)*f*arcsin(c*x)*g^2-3*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(
c^2*x^2-1)*arcsin(c*x)*x*f*g^2-3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^
2*x^2+1)^(1/2)-I)*f^2*g-3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1
)^(1/2)-I)*f*g^2+3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^2/d^2/(c^2*x^2
-1)*f^2*g-3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^3/d^2/(c^2*x^2-1)*f*g
^2+b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^3/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+3/2*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^
2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*arcsin(c*x)^2*f*g^2+b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^2/d^2/(c^2*x^2-1)*arcsin(c*x
)*x^2-b*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)*x*f^3-3*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1
)*arcsin(c*x)*f^2*g-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-
I)*f^3-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*g^3-b*(-
d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c/d^2/(c^2*x^2-1)*f^3+b*(-d*(c^2*x^2-1)
)^(1/2)*(-c^2*x^2+1)^(1/2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)/c^4/d^2/(c^2*x^2-1)*g^3+I*b*(-c^2*x^2+1)^(1/2)*(-d*(
c^2*x^2-1))^(1/2)/c/d^2/(c^2*x^2-1)*f^3*arcsin(c*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((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} +{\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*
arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^3*(b*arcsin(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)