3.53 \(\int \frac{(f+g x)^4 (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=528 \[ \frac{f g \left (1-c^2 x^2\right ) \left (2 c^2 f^2-5 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (2 c^2 f x \left (c^2 f^2-2 g^2\right )+g \left (c^2 f^2-3 g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x)^3 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x)^2 \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-2 g) (c f+g)^3 \log (1-c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 (c f+2 g) \log (c x+1)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

-(b*(f + g*x)^2*(c^2*f^2 + g^2 + 2*c^2*f*g*x))/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (b*f*g^3*x*
Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*c^5*d^2*Sqrt[d
- c^2*d*x^2]) + ((f + g*x)*(g*(c^2*f^2 - 3*g^2) + 2*c^2*f*(c^2*f^2 - 2*g^2)*x)*(a + b*ArcSin[c*x]))/(3*c^4*d^
2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(f + g*x)^3*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2
*d*x^2]) + (f*g*(2*c^2*f^2 - 5*g^2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (g^4*
Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*(c*f + g)^3*
Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*(c*f + 2*g)*Sqrt[1 - c^2*x^2]
*Log[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.752741, antiderivative size = 754, normalized size of antiderivative = 1.43, number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {4777, 739, 819, 641, 216, 4761, 774, 633, 31, 4641} \[ \frac{f g \left (1-c^2 x^2\right ) \left (2 c^2 f^2-5 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (2 c^2 f x \left (c^2 f^2-2 g^2\right )+g \left (c^2 f^2-3 g^2\right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x)^3 \left (c^2 f x+g\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f g x \sqrt{1-c^2 x^2} \left (2 c^2 f^2-5 g^2\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g x \sqrt{1-c^2 x^2} \left (c^2 f^2-2 g^2\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x)^2 \left (c^2 f^2+2 c^2 f g x+g^2\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g \sqrt{1-c^2 x^2} (c f+g)^3 \log (1-c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b g \sqrt{1-c^2 x^2} (c f-g)^3 \log (c x+1)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (2 c f-3 g) (c f+g)^3 \log (1-c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 (2 c f+3 g) \log (c x+1)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

-(b*(f + g*x)^2*(c^2*f^2 + g^2 + 2*c^2*f*g*x))/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (2*b*f*g^3*
x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f*g*(2*c^2*f^2 - 5*g^2)*x*Sqrt[1 - c^2*x^2])/(3*c^3*
d^2*Sqrt[d - c^2*d*x^2]) + (2*b*f*g*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (
b*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*c^5*d^2*Sqrt[d - c^2*d*x^2]) + ((f + g*x)*(g*(c^2*f^2 - 3*g^2) + 2*c
^2*f*(c^2*f^2 - 2*g^2)*x)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(f + g*x)^3*(a
+ b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f*g*(2*c^2*f^2 - 5*g^2)*(1 - c^2*x^2)*(a +
b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c^
5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(2*c*f - 3*g)*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^5*d^2*Sqrt[d -
c^2*d*x^2]) - (b*g*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^
3*g*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*(2*c*f + 3*g)*Sqrt[1 - c^
2*x^2]*Log[1 + c*x])/(6*c^5*d^2*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 739

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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 774

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/c, x] + Dist[1
/c, Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, 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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \left (\frac{f g \left (2 c^2 f^2-5 g^2\right )}{3 c^4}+\frac{\left (g+c^2 f x\right ) (f+g x)^3}{3 c^2 \left (1-c^2 x^2\right )^2}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right )}{3 c^4 \left (1-c^2 x^2\right )}+\frac{g^4 \sin ^{-1}(c x)}{c^5 \sqrt{1-c^2 x^2}}\right ) \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right )}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\left (g+c^2 f x\right ) (f+g x)^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b g^4 \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{-c^2 f g \left (c^2 f^2-3 g^2\right )-2 c^2 f g \left (c^2 f^2-2 g^2\right )-c^2 \left (g^2 \left (c^2 f^2-3 g^2\right )+2 c^2 f^2 \left (c^2 f^2-2 g^2\right )\right ) x}{1-c^2 x^2} \, dx}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{(f+g x) \left (2 g \left (c^2 f^2+g^2\right )+4 c^2 f g^2 x\right )}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{-4 c^2 f g^3-2 c^2 f g \left (c^2 f^2+g^2\right )-c^2 \left (4 c^2 f^2 g^2+2 g^2 \left (c^2 f^2+g^2\right )\right ) x}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (2 c f-3 g) (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{c-c^2 x} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 (2 c f+3 g) \sqrt{1-c^2 x^2}\right ) \int \frac{1}{-c-c^2 x} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (2 c f-3 g) (c f+g)^3 \sqrt{1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 (2 c f+3 g) \sqrt{1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 g \sqrt{1-c^2 x^2}\right ) \int \frac{1}{-c-c^2 x} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b g (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{c-c^2 x} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b (f+g x)^2 \left (c^2 f^2+g^2+2 c^2 f g x\right )}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f g \left (2 c^2 f^2-5 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left (g \left (c^2 f^2-3 g^2\right )+2 c^2 f \left (c^2 f^2-2 g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g+c^2 f x\right ) (f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f g \left (2 c^2 f^2-5 g^2\right ) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (2 c f-3 g) (c f+g)^3 \sqrt{1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g (c f+g)^3 \sqrt{1-c^2 x^2} \log (1-c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 g \sqrt{1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 (2 c f+3 g) \sqrt{1-c^2 x^2} \log (1+c x)}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 3.23103, size = 868, normalized size = 1.64 \[ \frac{b \left (4 c x \sin ^{-1}(c x)+\frac{\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-1}{\sqrt{1-c^2 x^2}}+4 \sqrt{1-c^2 x^2} \log \left (\sqrt{1-c^2 x^2}\right )\right ) f^4}{6 c d^2 \sqrt{d \left (1-c^2 x^2\right )}}+\frac{b g \left (8 \sin ^{-1}(c x)+\cos \left (3 \sin ^{-1}(c x)\right ) \left (\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )+3 \sqrt{1-c^2 x^2} \left (\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-2 \sin \left (2 \sin ^{-1}(c x)\right )\right ) f^3}{6 c^2 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\frac{b g^2 \left (-2 c x \sin ^{-1}(c x)+\frac{\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-1}{\sqrt{1-c^2 x^2}}-2 \sqrt{1-c^2 x^2} \log \left (\sqrt{1-c^2 x^2}\right )\right ) f^2}{c^3 d^2 \sqrt{d \left (1-c^2 x^2\right )}}-\frac{b g^3 \left (12 \cos \left (2 \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+4 \sin ^{-1}(c x)+5 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+15 \sqrt{1-c^2 x^2} \left (\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-5 \cos \left (3 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin \left (2 \sin ^{-1}(c x)\right )\right ) f}{6 c^4 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\sqrt{-d \left (c^2 x^2-1\right )} \left (\frac{a c^4 x f^4+4 a c^2 g f^3+6 a c^2 g^2 x f^2+4 a g^3 f+a g^4 x}{3 c^4 d^3 \left (c^2 x^2-1\right )^2}-\frac{2 a \left (c^4 x f^4-3 c^2 g^2 x f^2-6 g^3 f-2 g^4 x\right )}{3 c^4 d^3 \left (c^2 x^2-1\right )}\right )-\frac{a g^4 \tan ^{-1}\left (\frac{c x \sqrt{-d \left (c^2 x^2-1\right )}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{c^5 d^{5/2}}+\frac{b g^4 \left (\sqrt{1-c^2 x^2} \left (3 \sin ^{-1}(c x)^2-8 \log \left (\sqrt{1-c^2 x^2}\right )\right )-\frac{\frac{2 \sin ^{-1}(c x) \sin \left (3 \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+1}{\sqrt{1-c^2 x^2}}\right )}{6 c^5 d^2 \sqrt{d \left (1-c^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((4*a*c^2*f^3*g + 4*a*f*g^3 + a*c^4*f^4*x + 6*a*c^2*f^2*g^2*x + a*g^4*x)/(3*c^4*d^3*
(-1 + c^2*x^2)^2) - (2*a*(-6*f*g^3 + c^4*f^4*x - 3*c^2*f^2*g^2*x - 2*g^4*x))/(3*c^4*d^3*(-1 + c^2*x^2))) - (a*
g^4*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^5*d^(5/2)) + (b*f^2*g^2*(-2*c*x*ArcSi
n[c*x] + (-1 + (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2])/Sqrt[1 - c^2*x^2] - 2*Sqrt[1 - c^2*x^2]*Log[Sqrt[1 - c^2
*x^2]]))/(c^3*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b*f^4*(4*c*x*ArcSin[c*x] + (-1 + (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*
x^2])/Sqrt[1 - c^2*x^2] + 4*Sqrt[1 - c^2*x^2]*Log[Sqrt[1 - c^2*x^2]]))/(6*c*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b*f^
3*g*(8*ArcSin[c*x] + 3*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2
] + Sin[ArcSin[c*x]/2]]) + Cos[3*ArcSin[c*x]]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c
*x]/2] + Sin[ArcSin[c*x]/2]]) - 2*Sin[2*ArcSin[c*x]]))/(6*c^2*d*(d*(1 - c^2*x^2))^(3/2)) - (b*f*g^3*(4*ArcSin[
c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 5*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] +
15*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]
/2]]) - 5*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 2*Sin[2*ArcSin[c*x]]))/(6*c^4*d*(d
*(1 - c^2*x^2))^(3/2)) + (b*g^4*(Sqrt[1 - c^2*x^2]*(3*ArcSin[c*x]^2 - 8*Log[Sqrt[1 - c^2*x^2]]) - (1 + (2*ArcS
in[c*x]*Sin[3*ArcSin[c*x]])/Sqrt[1 - c^2*x^2])/Sqrt[1 - c^2*x^2]))/(6*c^5*d^2*Sqrt[d*(1 - c^2*x^2)])

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Maple [C]  time = 0.823, size = 6743, normalized size = 12.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)

[Out]

result too large to display

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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)^4*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),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^{4} x^{4} + 4 \, a f g^{3} x^{3} + 6 \, a f^{2} g^{2} x^{2} + 4 \, a f^{3} g x + a f^{4} +{\left (b g^{4} x^{4} + 4 \, b f g^{3} x^{3} + 6 \, b f^{2} g^{2} x^{2} + 4 \, b f^{3} g x + b f^{4}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(a*g^4*x^4 + 4*a*f*g^3*x^3 + 6*a*f^2*g^2*x^2 + 4*a*f^3*g*x + a*f^4 + (b*g^4*x^4 + 4*b*f*g^3*x^3 + 6*
b*f^2*g^2*x^2 + 4*b*f^3*g*x + b*f^4)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^
3*x^2 - d^3), 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)**4*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),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 )}^{4}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^4*(b*arcsin(c*x) + a)/(-c^2*d*x^2 + d)^(5/2), x)