3.104 \(\int \frac{(f+g x+h x^2) (a+b \sin ^{-1}(c x))}{(d+e x)^5} \, dx\)

Optimal. Leaf size=470 \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{b c \sqrt{1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c^3 \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \]

[Out]

(b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^3) - (b*c*(4*e^2*(e*g - 2*d*
h) - c^2*d*(5*e^2*f - d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c^2*d^2 - e^2)^2*(d + e*x)^2) + (b*c*(12*e
^4*h + c^4*d^2*(11*e^2*f + d*e*g - d^2*h) + 4*c^2*e^2*(e^2*f - 4*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c
^2*d^2 - e^2)^3*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x)^4) - ((e*g - 2*d*h
)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - (h*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - (b*c^3*(4*e^4*(e*g
- 5*d*h) - c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) - 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(24*e^3*(c^2*d^2 - e^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.943502, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{b c \sqrt{1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c^3 \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^3) - (b*c*(4*e^2*(e*g - 2*d*
h) - c^2*d*(5*e^2*f - d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c^2*d^2 - e^2)^2*(d + e*x)^2) + (b*c*(12*e
^4*h + c^4*d^2*(11*e^2*f + d*e*g - d^2*h) + 4*c^2*e^2*(e^2*f - 4*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c
^2*d^2 - e^2)^3*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x)^4) - ((e*g - 2*d*h
)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - (h*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - (b*c^3*(4*e^4*(e*g
- 5*d*h) - c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) - 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(24*e^3*(c^2*d^2 - e^2)^(7/2))

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)^5} \, dx &=-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-(b c) \int \frac{-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{12 e^3 (d+e x)^4 \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 )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{(b c) \int \frac{-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{(d+e x)^4 \sqrt{1-c^2 x^2}} \, dx}{12 e^3}\\ &=\frac{b c \left (e^2 f-d e g+d^2 h\right ) \sqrt{1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{(b c) \int \frac{3 \left (2 e^2 (2 e g-d h)-c^2 d \left (3 e^2 f+d e g+d^2 h\right )\right )+6 e \left (3 e^2 h+c^2 \left (e^2 f-d e g-2 d^2 h\right )\right ) x}{(d+e x)^3 \sqrt{1-c^2 x^2}} \, dx}{36 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{(b c) \int \frac{-6 \left (6 e^4 h+2 c^2 e^2 \left (e^2 f-3 d e g-d^2 h\right )+c^4 d^2 \left (3 e^2 f+d e g+d^2 h\right )\right )-3 c^2 e \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{72 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{24 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac{\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+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 )}{24 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}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac{b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac{b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt{1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac{\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac{h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac{b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+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 )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 2.93473, size = 575, normalized size = 1.22 \[ -\frac{\frac{6 a \left (d^2 h-d e g+e^2 f\right )}{(d+e x)^4}+\frac{8 a (e g-2 d h)}{(d+e x)^3}+\frac{12 a h}{(d+e x)^2}+\frac{b c e \sqrt{1-c^2 x^2} \left (c^4 d^2 \left (d^2 e^2 (18 f+x (g-h x))-d^3 e (2 g+5 h x)-2 d^4 h+d e^3 x (27 f+g x)+11 e^4 f x^2\right )+c^2 e^2 \left (d^2 e^2 (x (4 h x-35 g)-5 f)+d^3 e (19 h x-15 g)+11 d^4 h+d e^3 x (3 f-16 g x)+4 e^4 f x^2\right )+2 e^4 \left (3 d^2 h+d e (g+8 h x)+e^2 (f+2 x (g+3 h x))\right )\right )}{\left (e^2-c^2 d^2\right )^3 (d+e x)^3}+\frac{b c^3 \log \left (\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right ) \left (2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )+c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )-4 e^4 (e g-5 d h)\right )}{(c d-e)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}-\frac{b c^3 \log (d+e x) \left (2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )+c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )-4 e^4 (e g-5 d h)\right )}{(c d-e)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}+\frac{2 b \sin ^{-1}(c x) \left (d^2 h+d e (g+4 h x)+e^2 \left (3 f+4 g x+6 h x^2\right )\right )}{(d+e x)^4}}{24 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^4 + (8*a*(e*g - 2*d*h))/(d + e*x)^3 + (12*a*h)/(d + e*x)^2 + (b*c*e*
Sqrt[1 - c^2*x^2]*(c^4*d^2*(-2*d^4*h + 11*e^4*f*x^2 + d*e^3*x*(27*f + g*x) - d^3*e*(2*g + 5*h*x) + d^2*e^2*(18
*f + x*(g - h*x))) + 2*e^4*(3*d^2*h + d*e*(g + 8*h*x) + e^2*(f + 2*x*(g + 3*h*x))) + c^2*e^2*(11*d^4*h + 4*e^4
*f*x^2 + d*e^3*x*(3*f - 16*g*x) + d^3*e*(-15*g + 19*h*x) + d^2*e^2*(-5*f + x*(-35*g + 4*h*x)))))/((-(c^2*d^2)
+ e^2)^3*(d + e*x)^3) + (2*b*(d^2*h + d*e*(g + 4*h*x) + e^2*(3*f + 4*g*x + 6*h*x^2))*ArcSin[c*x])/(d + e*x)^4
- (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h
))*Log[d + e*x])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(
9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + 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)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]))/(24*e^3)

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Maple [B]  time = 0.013, size = 3005, normalized size = 6.4 \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)^5,x)

[Out]

2/3*c^4*b/e^2*g*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)-7/8
*c^5*b/e^3*g*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))-1/12*c^4*b/
e^4/(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*g-5/24*c^5*b/e^
3*d^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)*g-1/4*c^4*a/e
^3/(c*e*x+c*d)^4*d^2*h+2/3*c^3*a/e^3/(c*e*x+c*d)^3*d*h-1/2*c^2*b*arcsin(c*x)*h/e^3/(c*e*x+c*d)^2-1/2*c^2*a*h/e
^3/(c*e*x+c*d)^2-1/3*c^3*b/e^4/(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)*d*h+5/24*c^5*b/e^4*d^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)*h+1/12*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^2*h-5/8*c^7*b/e^4*d^5/(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+5/8*c^6*b/e^3*d^4/(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+11/8*c^5*b/e^4*d^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))*h+
5/24*c^5*b/e^2*d/(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-
5/8*c^6*b/e^2*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)*g+5
/8*c^6*b/e*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)*f+5/8*
c^7*b/e^3*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))*g-5/6*c^3*b/e^
4/(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))*d*h-7/6*c^4*b/e^3*d^2/(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)*h-5/8*c^7*b/e^2*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))*f+3/8*c^5*b/e^2*d/(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-1/4*c^4*b*arcsin(c*x)/e/(c*e*x+c*d)^4*f-1/3
*c^3*b*arcsin(c*x)*g/e^2/(c*e*x+c*d)^3+1/4*c^4*a/e^2/(c*e*x+c*d)^4*d*g+1/2*c^2*b/e^3*h/(c^2*d^2-e^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)-1/4*c^4*b*arcsin(c*x)/e^3/(c*e*x+c*d)^4*d^2*h+
2/3*c^3*b*arcsin(c*x)/e^3/(c*e*x+c*d)^3*d*h-1/3*c^3*a*g/e^2/(c*e*x+c*d)^3-1/4*c^4*a/e/(c*e*x+c*d)^4*f+1/4*c^4*
b*arcsin(c*x)/e^2/(c*e*x+c*d)^4*d*g+1/6*c^3*b/e^3*g/(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)+1/6*c^3*b/e^3*g/(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))+1/12*c^4*b/e^3/(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)*f-1/6*c^4*b/e/(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)*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (4 \, e x + d\right )} a g}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac{{\left (6 \, e^{2} x^{2} + 4 \, d e x + d^{2}\right )} a h}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} - \frac{a f}{4 \,{\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac{{\left (6 \, b e^{2} h x^{2} + 3 \, b e^{2} f + b d e g + b d^{2} h + 4 \,{\left (b e^{2} g + b d e h\right )} x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )} \int \frac{{\left (6 \, b c e^{2} h x^{2} + 3 \, b c e^{2} f + b c d e g + b c d^{2} h + 4 \,{\left (b c e^{2} g + b c d e h\right )} x\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^{8} + 4 \, c^{4} d e^{6} x^{7} - 4 \, c^{2} d^{3} e^{4} x^{3} - c^{2} d^{4} e^{3} x^{2} +{\left (6 \, c^{4} d^{2} e^{5} - c^{2} e^{7}\right )} x^{6} + 4 \,{\left (c^{4} d^{3} e^{4} - c^{2} d e^{6}\right )} x^{5} +{\left (c^{4} d^{4} e^{3} - 6 \, c^{2} d^{2} e^{5}\right )} x^{4} -{\left (c^{2} e^{7} x^{6} + 4 \, c^{2} d e^{6} x^{5} - 4 \, d^{3} e^{4} x - d^{4} e^{3} +{\left (6 \, c^{2} d^{2} e^{5} - e^{7}\right )} x^{4} + 4 \,{\left (c^{2} d^{3} e^{4} - d e^{6}\right )} x^{3} +{\left (c^{2} d^{4} e^{3} - 6 \, d^{2} e^{5}\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \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)^5,x, algorithm="maxima")

[Out]

-1/12*(4*e*x + d)*a*g/(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x + d^4*e^2) - 1/12*(6*e^2*x^2 + 4*d*
e*x + d^2)*a*h/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3) - 1/4*a*f/(e^5*x^4 + 4*d*e^4*x^
3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) - 1/12*((6*b*e^2*h*x^2 + 3*b*e^2*f + b*d*e*g + b*d^2*h + 4*(b*e^2*g +
b*d*e*h)*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 12*(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^
4*x + d^4*e^3)*integrate(1/12*(6*b*c*e^2*h*x^2 + 3*b*c*e^2*f + b*c*d*e*g + b*c*d^2*h + 4*(b*c*e^2*g + b*c*d*e*
h)*x)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))/(c^4*e^7*x^8 + 4*c^4*d*e^6*x^7 - 4*c^2*d^3*e^4*x^3 - c^2*d^4*e^
3*x^2 + (6*c^4*d^2*e^5 - c^2*e^7)*x^6 + 4*(c^4*d^3*e^4 - c^2*d*e^6)*x^5 + (c^4*d^4*e^3 - 6*c^2*d^2*e^5)*x^4 +
(c^2*e^7*x^6 + 4*c^2*d*e^6*x^5 - 4*d^3*e^4*x - d^4*e^3 + (6*c^2*d^2*e^5 - e^7)*x^4 + 4*(c^2*d^3*e^4 - d*e^6)*x
^3 + (c^2*d^4*e^3 - 6*d^2*e^5)*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x))/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*
x^2 + 4*d^3*e^4*x + d^4*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)^5,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 )^{5}}\, 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)**5,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x + h*x**2)/(d + e*x)**5, 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 )}^{5}}\,{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)^5,x, algorithm="giac")

[Out]

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