3.96 \(\int \frac{(f+g x) (a+b \sin ^{-1}(c x))}{(d+e x)^6} \, dx\)

Optimal. Leaf size=457 \[ -\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac{b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+2 c^4 d^4 (d g+4 e f)+3 e^4 (e f-6 d g)\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}} \]

[Out]

(b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(20*e*(c^2*d^2 - e^2)*(d + e*x)^4) - (b*c*(5*e^2*g - c^2*d*(7*e*f - 2*d*g)
)*Sqrt[1 - c^2*x^2])/(60*e*(c^2*d^2 - e^2)^2*(d + e*x)^3) + (b*c^3*(e^2*(9*e*f - 34*d*g) + c^2*d^2*(26*e*f - d
*g))*Sqrt[1 - c^2*x^2])/(120*e*(c^2*d^2 - e^2)^3*(d + e*x)^2) - (b*c^3*(4*e^4*g - c^2*d*e^2*(11*e*f - 18*d*g)
- c^4*d^3*(10*e*f + d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^4*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c
*x]))/(5*e^2*(d + e*x)^5) - (g*(a + b*ArcSin[c*x]))/(4*e^2*(d + e*x)^4) + (b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g
) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]
)])/(40*e^2*(c^2*d^2 - e^2)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.965565, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 4753, 12, 835, 807, 725, 204} \[ -\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac{b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+2 c^4 d^4 (d g+4 e f)+3 e^4 (e f-6 d g)\right ) \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

(b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(20*e*(c^2*d^2 - e^2)*(d + e*x)^4) - (b*c*(5*e^2*g - c^2*d*(7*e*f - 2*d*g)
)*Sqrt[1 - c^2*x^2])/(60*e*(c^2*d^2 - e^2)^2*(d + e*x)^3) + (b*c^3*(e^2*(9*e*f - 34*d*g) + c^2*d^2*(26*e*f - d
*g))*Sqrt[1 - c^2*x^2])/(120*e*(c^2*d^2 - e^2)^3*(d + e*x)^2) - (b*c^3*(4*e^4*g - c^2*d*e^2*(11*e*f - 18*d*g)
- c^4*d^3*(10*e*f + d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^4*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c
*x]))/(5*e^2*(d + e*x)^5) - (g*(a + b*ArcSin[c*x]))/(4*e^2*(d + e*x)^4) + (b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g
) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]
)])/(40*e^2*(c^2*d^2 - e^2)^(9/2))

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])

Rule 4753

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 12

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

Rule 835

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

Rule 807

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^6} \, dx &=-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-(b c) \int \frac{-4 e f-d g-5 e g x}{20 e^2 (d+e x)^5 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{(b c) \int \frac{-4 e f-d g-5 e g x}{(d+e x)^5 \sqrt{1-c^2 x^2}} \, dx}{20 e^2}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{(b c) \int \frac{4 \left (5 e^2 g-c^2 d (4 e f+d g)\right )+12 c^2 e (e f-d g) x}{(d+e x)^4 \sqrt{1-c^2 x^2}} \, dx}{80 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt{1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{(b c) \int \frac{-12 c^2 \left (e^2 (3 e f-8 d g)+c^2 d^2 (4 e f+d g)\right )-8 c^2 e \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) x}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{240 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt{1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt{1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{(b c) \int \frac{8 c^2 \left (10 e^4 g-c^2 d e^2 (23 e f-28 d g)-3 c^4 d^3 (4 e f+d g)\right )+4 c^4 e \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{480 e^2 \left (c^2 d^2-e^2\right )^3}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt{1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt{1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}+\frac{\left (b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{40 e^2 \left (c^2 d^2-e^2\right )^4}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt{1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt{1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac{\left (b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac{e+c^2 d x}{\sqrt{1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^4}\\ &=\frac{b c (e f-d g) \sqrt{1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt{1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt{1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac{g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}+\frac{b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.39366, size = 494, normalized size = 1.08 \[ \frac{\frac{3 a (8 d g-8 e f)}{(d+e x)^5}-\frac{30 a g}{(d+e x)^4}+\frac{b c e \sqrt{1-c^2 x^2} \left (-2 \left (e^2-c^2 d^2\right )^2 (d+e x) \left (c^2 d (2 d g-7 e f)+5 e^2 g\right )+5 c^2 (d+e x)^3 \left (c^4 d^3 (d g+10 e f)+c^2 d e^2 (11 e f-18 d g)-4 e^4 g\right )-c^2 \left (c^2 d^2-e^2\right ) (d+e x)^2 \left (c^2 d^2 (d g-26 e f)+e^2 (34 d g-9 e f)\right )-6 \left (e^2-c^2 d^2\right )^3 (e f-d g)\right )}{\left (e^2-c^2 d^2\right )^4 (d+e x)^4}-\frac{3 b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+2 c^4 d^4 (d g+4 e f)+3 e^4 (e f-6 d g)\right ) \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right )}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}+\frac{3 b c^5 \log (d+e x) \left (c^2 d^2 e^2 (24 e f-19 d g)+2 c^4 d^4 (d g+4 e f)+3 e^4 (e f-6 d g)\right )}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}-\frac{6 b \sin ^{-1}(c x) (d g+4 e f+5 e g x)}{(d+e x)^5}}{120 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

((3*a*(-8*e*f + 8*d*g))/(d + e*x)^5 - (30*a*g)/(d + e*x)^4 + (b*c*e*Sqrt[1 - c^2*x^2]*(-6*(-(c^2*d^2) + e^2)^3
*(e*f - d*g) - 2*(-(c^2*d^2) + e^2)^2*(5*e^2*g + c^2*d*(-7*e*f + 2*d*g))*(d + e*x) - c^2*(c^2*d^2 - e^2)*(c^2*
d^2*(-26*e*f + d*g) + e^2*(-9*e*f + 34*d*g))*(d + e*x)^2 + 5*c^2*(-4*e^4*g + c^2*d*e^2*(11*e*f - 18*d*g) + c^4
*d^3*(10*e*f + d*g))*(d + e*x)^3))/((-(c^2*d^2) + e^2)^4*(d + e*x)^4) - (6*b*(4*e*f + d*g + 5*e*g*x)*ArcSin[c*
x])/(d + e*x)^5 + (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log
[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]) - (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e
^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c
*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]))/(120*e^2)

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 2431, normalized size = 5.3 \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)*(a+b*arcsin(c*x))/(e*x+d)^6,x)

[Out]

-1/4*c^4*b*arcsin(c*x)*g/e^2/(c*e*x+c*d)^4+1/5*c^5*a/e^2/(c*e*x+c*d)^5*d*g-11/24*c^6*b/e*d/(c^2*d^2-e^2)^3/(c*
x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/8*c^9*b/e^2*d^4/(c^2*d^2-e^2)^4/(-(c
^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)
^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*f+7/8*c^9*b/e^3*d^5/(c^2*d^2-e^2)^4/(-(c^2*d^2-e
^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c
/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*g+9/20*c^5*b/e^3/(c^2*d^2-e^2)^2/(-(c^2*d^2-e^2)/e^2)^(1
/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c
/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*d*g+13/12*c^6*b/e^2*d^2/(c^2*d^2-e^2)^3/(c*x+d*c/e)*(-(c*x+d*c/e)^2
+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g-7/60*c^6*b/e^4*d^2/(c^2*d^2-e^2)^2/(c*x+d*c/e)^3*(-(c*x+d*c/e)
^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/60*c^6*b/e^3*d/(c^2*d^2-e^2)^2/(c*x+d*c/e)^3*(-(c*x+d*c/e)
^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/24*c^7*b/e^3*d^3/(c^2*d^2-e^2)^3/(c*x+d*c/e)^2*(-(c*x+d*c/
e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/24*c^7*b/e^2*d^2/(c^2*d^2-e^2)^3/(c*x+d*c/e)^2*(-(c*x+d*
c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f+3/4*c^7*b/e^2*d^2/(c^2*d^2-e^2)^3/(-(c^2*d^2-e^2)/e^2)^(
1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*
c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*f-11/8*c^7*b/e^3*d^3/(c^2*d^2-e^2)^3/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln
((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c
^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*g+7/8*c^8*b/e*d^3/(c^2*d^2-e^2)^4/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*
x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/8*c^8*b/e^2*d^4/(c^2*d^2-e^2)^4/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x
+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g-1/20*c^5*b/e^5/(c^2*d^2-e^2)/(c*x+d*c/e)^4*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c
/e)-(c^2*d^2-e^2)/e^2)^(1/2)*d*g+17/60*c^5*b/e^3/(c^2*d^2-e^2)^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*
c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*d*g-1/5*c^5*a/e/(c*e*x+c*d)^5*f-1/5*c^5*b*arcsin(c*x)/e/(c*e*x+c*d)^5*f-1/4*c^4*
a*g/e^2/(c*e*x+c*d)^4-1/6*c^4*b/e^2*g/(c^2*d^2-e^2)^2/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2
-e^2)/e^2)^(1/2)-3/40*c^5*b/e^2/(c^2*d^2-e^2)^2/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c
*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/
e))*f+1/5*c^5*b*arcsin(c*x)/e^2/(c*e*x+c*d)^5*d*g+1/20*c^5*b/e^4/(c^2*d^2-e^2)/(c*x+d*c/e)^4*(-(c*x+d*c/e)^2+2
*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-3/40*c^5*b/e^2/(c^2*d^2-e^2)^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d
*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f+1/12*c^4*b/e^4*g/(c^2*d^2-e^2)/(c*x+d*c/e)^3*(-(c*x+d*c/e)^2+2*d*c
/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (5 \, e x + d\right )} a g}{20 \,{\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} - \frac{a f}{5 \,{\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac{{\left (5 \, b e g x + 4 \, b e f + b d g\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )} \int \frac{{\left (5 \, b c e g x + 4 \, b c e f + b c d g\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{4} e^{7} x^{9} + 5 \, c^{4} d e^{6} x^{8} - 5 \, c^{2} d^{4} e^{3} x^{3} - c^{2} d^{5} e^{2} x^{2} +{\left (10 \, c^{4} d^{2} e^{5} - c^{2} e^{7}\right )} x^{7} + 5 \,{\left (2 \, c^{4} d^{3} e^{4} - c^{2} d e^{6}\right )} x^{6} + 5 \,{\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5}\right )} x^{5} +{\left (c^{4} d^{5} e^{2} - 10 \, c^{2} d^{3} e^{4}\right )} x^{4} -{\left (c^{2} e^{7} x^{7} + 5 \, c^{2} d e^{6} x^{6} - 5 \, d^{4} e^{3} x - d^{5} e^{2} +{\left (10 \, c^{2} d^{2} e^{5} - e^{7}\right )} x^{5} + 5 \,{\left (2 \, c^{2} d^{3} e^{4} - d e^{6}\right )} x^{4} + 5 \,{\left (c^{2} d^{4} e^{3} - 2 \, d^{2} e^{5}\right )} x^{3} +{\left (c^{2} d^{5} e^{2} - 10 \, d^{3} e^{4}\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}}{20 \,{\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(5*e*x + d)*a*g/(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e^4*x^2 + 5*d^4*e^3*x + d^5*e^2) - 1/5*
a*f/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 1/20*((5*b*e*g*x + 4*b*e
*f + b*d*g)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 20*(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e
^4*x^2 + 5*d^4*e^3*x + d^5*e^2)*integrate(1/20*(5*b*c*e*g*x + 4*b*c*e*f + b*c*d*g)*e^(1/2*log(c*x + 1) + 1/2*l
og(-c*x + 1))/(c^4*e^7*x^9 + 5*c^4*d*e^6*x^8 - 5*c^2*d^4*e^3*x^3 - c^2*d^5*e^2*x^2 + (10*c^4*d^2*e^5 - c^2*e^7
)*x^7 + 5*(2*c^4*d^3*e^4 - c^2*d*e^6)*x^6 + 5*(c^4*d^4*e^3 - 2*c^2*d^2*e^5)*x^5 + (c^4*d^5*e^2 - 10*c^2*d^3*e^
4)*x^4 + (c^2*e^7*x^7 + 5*c^2*d*e^6*x^6 - 5*d^4*e^3*x - d^5*e^2 + (10*c^2*d^2*e^5 - e^7)*x^5 + 5*(2*c^2*d^3*e^
4 - d*e^6)*x^4 + 5*(c^2*d^4*e^3 - 2*d^2*e^5)*x^3 + (c^2*d^5*e^2 - 10*d^3*e^4)*x^2)*e^(log(c*x + 1) + log(-c*x
+ 1))), x))/(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e^4*x^2 + 5*d^4*e^3*x + d^5*e^2)

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Fricas [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((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="fricas")

[Out]

Timed out

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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((g*x+f)*(a+b*asin(c*x))/(e*x+d)**6,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^6, x)