3.105 \(\int \frac{(f+g x+h x^2) (a+b \sin ^{-1}(c x))}{(d+e x)^6} \, dx\)

Optimal. Leaf size=593 \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )+c^4 d^3 (d g+10 e f)-4 e^3 (e g-5 d h)\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} \left (c^4 d^2 \left (-4 d^2 h-d e g+26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (5 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-2 d e g+7 e^2 f\right )\right )}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac{b c^3 \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )+3 c^4 d^2 e^2 \left (-6 d^2 h-19 d e g+24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.25549, antiderivative size = 593, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {698, 4753, 12, 1651, 835, 807, 725, 204} \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac{b c^3 \sqrt{1-c^2 x^2} \left (c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )+c^4 d^3 (d g+10 e f)-4 e^3 (e g-5 d h)\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} \left (c^4 d^2 \left (-4 d^2 h-d e g+26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left (5 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-2 d e g+7 e^2 f\right )\right )}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac{b c^3 \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )+3 c^4 d^2 e^2 \left (-6 d^2 h-19 d e g+24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

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{\left (f+g x+h x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^6} \, dx &=-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-(b c) \int \frac{-12 e^2 f-3 d e g-2 d^2 h-5 e (3 e g+2 d h) x-20 e^2 h x^2}{60 e^3 (d+e x)^5 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(b c) \int \frac{-12 e^2 f-3 d e g-2 d^2 h-5 e (3 e g+2 d h) x-20 e^2 h x^2}{(d+e x)^5 \sqrt{1-c^2 x^2}} \, dx}{60 e^3}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(b c) \int \frac{4 \left (5 e^2 (3 e g-2 d h)-c^2 d \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )+4 e \left (20 e^2 h+c^2 \left (9 e^2 f-9 d e g-11 d^2 h\right )\right ) x}{(d+e x)^4 \sqrt{1-c^2 x^2}} \, dx}{240 e^3 \left (c^2 d^2-e^2\right )}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(b c) \int \frac{-12 \left (20 e^4 h+c^2 e^2 \left (9 e^2 f-24 d e g-d^2 h\right )+c^4 d^2 \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )-24 c^2 e \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) x}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{720 e^3 \left (c^2 d^2-e^2\right )^2}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{(b c) \int \frac{24 c^2 \left (10 e^4 (e g-4 d h)-c^2 d e^2 \left (23 e^2 f-28 d e g-7 d^2 h\right )-c^4 d^3 \left (12 e^2 f+3 d e g+2 d^2 h\right )\right )+12 c^2 e \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{1440 e^3 \left (c^2 d^2-e^2\right )^3}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac{b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac{\left (b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{120 e^3 \left (c^2 d^2-e^2\right )^4}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac{b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{\left (b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\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 )}{120 e^3 \left (c^2 d^2-e^2\right )^4}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{20 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac{b c \left (5 e^2 (e g-2 d h)-c^2 d \left (7 e^2 f-2 d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{60 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac{b c \left (20 e^4 h+c^4 d^2 \left (26 e^2 f-d e g-4 d^2 h\right )+c^2 e^2 \left (9 e^2 f-34 d e g+19 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{120 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)^2}+\frac{b c^3 \left (c^4 d^3 (10 e f+d g)-4 e^3 (e g-5 d h)+c^2 d e \left (11 e^2 f-18 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}+\frac{b c^3 \left (20 e^6 h+3 c^4 d^2 e^2 \left (24 e^2 f-19 d e g-6 d^2 h\right )+2 c^6 d^4 \left (12 e^2 f+3 d e g+2 d^2 h\right )+9 c^2 e^4 \left (e^2 f-6 d e g+11 d^2 h\right )\right ) \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{120 e^3 \left (c^2 d^2-e^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.66865, size = 682, normalized size = 1.15 \[ -\frac{\frac{24 a \left (d^2 h-d e g+e^2 f\right )}{(d+e x)^5}+\frac{30 a (e g-2 d h)}{(d+e x)^4}+\frac{40 a h}{(d+e x)^3}-\frac{b c e \sqrt{1-c^2 x^2} \left (5 c^2 e (d+e x)^3 \left (c^2 d e \left (d^2 h-18 d e g+11 e^2 f\right )+c^4 d^3 (d g+10 e f)-4 e^3 (e g-5 d h)\right )-\left (e^2-c^2 d^2\right ) (d+e x)^2 \left (c^4 \left (-d^2\right ) \left (4 d^2 h+d e g-26 e^2 f\right )+c^2 e^2 \left (19 d^2 h-34 d e g+9 e^2 f\right )+20 e^4 h\right )-2 \left (e^2-c^2 d^2\right )^2 (d+e x) \left (c^2 d \left (3 d^2 h+2 d e g-7 e^2 f\right )+5 e^2 (e g-2 d h)\right )+6 \left (c^2 d^2-e^2\right )^3 \left (d^2 h-d e g+e^2 f\right )\right )}{\left (e^2-c^2 d^2\right )^4 (d+e x)^4}+\frac{b c^3 \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right ) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )-3 c^4 d^2 e^2 \left (6 d^2 h+19 d e g-24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}-\frac{b c^3 \log (d+e x) \left (2 c^6 d^4 \left (2 d^2 h+3 d e g+12 e^2 f\right )-3 c^4 d^2 e^2 \left (6 d^2 h+19 d e g-24 e^2 f\right )+9 c^2 e^4 \left (11 d^2 h-6 d e g+e^2 f\right )+20 e^6 h\right )}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}+\frac{2 b \sin ^{-1}(c x) \left (2 d^2 h+d e (3 g+10 h x)+e^2 (12 f+5 x (3 g+4 h x))\right )}{(d+e x)^5}}{120 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

-((24*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^5 + (30*a*(e*g - 2*d*h))/(d + e*x)^4 + (40*a*h)/(d + e*x)^3 - (b*c*
e*Sqrt[1 - c^2*x^2]*(6*(c^2*d^2 - e^2)^3*(e^2*f - d*e*g + d^2*h) - 2*(-(c^2*d^2) + e^2)^2*(5*e^2*(e*g - 2*d*h)
+ c^2*d*(-7*e^2*f + 2*d*e*g + 3*d^2*h))*(d + e*x) - (-(c^2*d^2) + e^2)*(20*e^4*h - c^4*d^2*(-26*e^2*f + d*e*g
+ 4*d^2*h) + c^2*e^2*(9*e^2*f - 34*d*e*g + 19*d^2*h))*(d + e*x)^2 + 5*c^2*e*(c^4*d^3*(10*e*f + d*g) - 4*e^3*(
e*g - 5*d*h) + c^2*d*e*(11*e^2*f - 18*d*e*g + d^2*h))*(d + e*x)^3))/((-(c^2*d^2) + e^2)^4*(d + e*x)^4) + (2*b*
(2*d^2*h + d*e*(3*g + 10*h*x) + e^2*(12*f + 5*x*(3*g + 4*h*x)))*ArcSin[c*x])/(d + e*x)^5 - (b*c^3*(20*e^6*h +
2*c^6*d^4*(12*e^2*f + 3*d*e*g + 2*d^2*h) - 3*c^4*d^2*e^2*(-24*e^2*f + 19*d*e*g + 6*d^2*h) + 9*c^2*e^4*(e^2*f -
6*d*e*g + 11*d^2*h))*Log[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(20*e^6*h + 2
*c^6*d^4*(12*e^2*f + 3*d*e*g + 2*d^2*h) - 3*c^4*d^2*e^2*(-24*e^2*f + 19*d*e*g + 6*d^2*h) + 9*c^2*e^4*(e^2*f -
6*d*e*g + 11*d^2*h))*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^3)

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 4077, normalized size = 6.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x^2+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+7/24*c^7*b/e^4*d^4/(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)*h+5/6*c^4*b/e^3*h*d/(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)-59/120*c^5*b/e^4/(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^2*h-1/6*c^4*b/e^5/(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)*d*h+1/20*c^5*b/e^6/(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^2*h-41/24*c^6*b/e^3*d^3/(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)*h+7/8*c^8*b/e^3*d^5/(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)*h+2*c^7*b/e^4*d^4/(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))*h-53/40*c^5*b/e^4*h*d^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))-7/8*c^9*b/e^4*d^6/(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))*h+7/60*c^6*b/e^5*d^3/(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)*h-1/3*c^3*a*h/e^3/(c*e*x+c*d)^3-7/8*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+
1/2*c^4*a/e^3/(c*e*x+c*d)^4*d*h-1/5*c^5*a/e^3/(c*e*x+c*d)^5*d^2*h-1/3*c^3*b*arcsin(c*x)*h/e^3/(c*e*x+c*d)^3+1/
6*c^3*b/e^4*h/(c^2*d^2-e^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))-1/5*c^5*b*arcsin
(c*x)/e^3/(c*e*x+c*d)^5*d^2*h+1/6*c^3*b/e^4*h/(c^2*d^2-e^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)+1/2*c^4*b*arcsin(c*x)/e^3/(c*e*x+c*d)^4*d*h-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*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^2+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) - 1/30
*(10*e^2*x^2 + 5*d*e*x + d^2)*a*h/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5
*e^3) - 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) - 1/60*((20*b*
e^2*h*x^2 + 12*b*e^2*f + 3*b*d*e*g + 2*b*d^2*h + 5*(3*b*e^2*g + 2*b*d*e*h)*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(
-c*x + 1)) + 60*(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)*integrate(1/
60*(20*b*c*e^2*h*x^2 + 12*b*c*e^2*f + 3*b*c*d*e*g + 2*b*c*d^2*h + 5*(3*b*c*e^2*g + 2*b*c*d*e*h)*x)*e^(1/2*log(
c*x + 1) + 1/2*log(-c*x + 1))/(c^4*e^8*x^9 + 5*c^4*d*e^7*x^8 - 5*c^2*d^4*e^4*x^3 - c^2*d^5*e^3*x^2 + (10*c^4*d
^2*e^6 - c^2*e^8)*x^7 + 5*(2*c^4*d^3*e^5 - c^2*d*e^7)*x^6 + 5*(c^4*d^4*e^4 - 2*c^2*d^2*e^6)*x^5 + (c^4*d^5*e^3
- 10*c^2*d^3*e^5)*x^4 + (c^2*e^8*x^7 + 5*c^2*d*e^7*x^6 - 5*d^4*e^4*x - d^5*e^3 + (10*c^2*d^2*e^6 - e^8)*x^5 +
5*(2*c^2*d^3*e^5 - d*e^7)*x^4 + 5*(c^2*d^4*e^4 - 2*d^2*e^6)*x^3 + (c^2*d^5*e^3 - 10*d^3*e^5)*x^2)*e^(log(c*x
+ 1) + log(-c*x + 1))), x))/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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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((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**6,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x + h*x**2)/(d + e*x)**6, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (h x^{2} + 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((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="giac")

[Out]

integrate((h*x^2 + g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^6, x)