3.112 \(\int \frac {(f+g x+h x^2+i x^3) (a+b \sin ^{-1}(c x))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=1278 \[ -\frac {11 b c^3 i \sqrt {1-c^2 x^2} d^4}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {11 b c^3 \left (2 c^2 d^2+e^2\right ) i \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^3}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {11 b c i \sqrt {1-c^2 x^2} d^3}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (4 c^2 h d^2+e (2 e h+81 d i)\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (d (2 e h+9 d i) c^2+18 e^2 i\right ) \sqrt {1-c^2 x^2} d^2}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c (2 e h+27 d i) \sqrt {1-c^2 x^2} d^2}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 d^2 g c^4+\left (-18 i d^2-18 e h d+e^2 g\right ) c^2-36 e^2 i\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c \left (4 e^2 (e h+6 d i)-c^2 d \left (6 i d^2-2 e h d+e^2 g\right )\right ) \sqrt {1-c^2 x^2} d}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (-18 i d^2-6 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {i b i \sin ^{-1}(c x)^2}{2 e^4}-\frac {(e h-3 d i) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}+\frac {b c \left (4 d^2 f c^4+\left (6 h d^2-9 e g d+2 e^2 f\right ) c^2+12 e^2 h\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b i \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b i \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b i \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {i \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {i b i \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b i \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (-2 h d^2-e g d+2 e^2 f\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (6 h d^2-3 e g d+2 e^2 f\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2} \]

[Out]

-1/2*I*b*i*arcsin(c*x)^2/e^4-1/3*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))/e^4/(e*x+d)^3-1/2*(3*d^2*i-2
*d*e*h+e^2*g)*(a+b*arcsin(c*x))/e^4/(e*x+d)^2-(-3*d*i+e*h)*(a+b*arcsin(c*x))/e^4/(e*x+d)+1/12*b*c*(4*c^4*d^2*f
+12*e^2*h+c^2*(6*d^2*h-9*d*e*g+2*e^2*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e/(c^2*d^2
-e^2)^(5/2)-11/12*b*c^3*d^3*(2*c^2*d^2+e^2)*i*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(
c^2*d^2-e^2)^(5/2)+1/12*b*c^3*d^2*(4*c^2*d^2*h+e*(81*d*i+2*e*h))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*
x^2+1)^(1/2))/e^3/(c^2*d^2-e^2)^(5/2)+1/12*b*c*d*(2*c^4*d^2*g-36*e^2*i+c^2*(-18*d^2*i-18*d*e*h+e^2*g))*arctan(
(c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^2/(c^2*d^2-e^2)^(5/2)-b*i*arcsin(c*x)*ln(e*x+d)/e^4+i*(a
+b*arcsin(c*x))*ln(e*x+d)/e^4+b*i*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e
^4+b*i*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4-I*b*i*polylog(2,I*e*(I*c
*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4-I*b*i*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2
*d^2-e^2)^(1/2)))/e^4+1/12*b*c*(6*d^2*h-3*d*e*g+2*e^2*f)*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)/(e*x+d)^2-11/12*
b*c*d^3*i*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)^2+1/12*b*c*d^2*(27*d*i+2*e*h)*(-c^2*x^2+1)^(1/2)/e^3/(c
^2*d^2-e^2)/(e*x+d)^2+1/12*b*c*d*(-18*d^2*i-6*d*e*h+e^2*g)*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)/(e*x+d)^2-1/4*
b*c*(2*e^2*(-4*d*h+e*g)-c^2*d*(-2*d^2*h-d*e*g+2*e^2*f))*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)^2/(e*x+d)-11/4*b*
c^3*d^4*i*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^2)^2/(e*x+d)+1/4*b*c*d^2*(18*e^2*i+c^2*d*(9*d*i+2*e*h))*(-c^2*x^2+
1)^(1/2)/e^3/(c^2*d^2-e^2)^2/(e*x+d)-1/4*b*c*d*(4*e^2*(6*d*i+e*h)-c^2*d*(6*d^2*i-2*d*e*h+e^2*g))*(-c^2*x^2+1)^
(1/2)/e^3/(c^2*d^2-e^2)^2/(e*x+d)

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Rubi [A]  time = 2.86, antiderivative size = 1278, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {1850, 4753, 12, 6742, 745, 807, 725, 204, 835, 1651, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac {11 b c^3 i \sqrt {1-c^2 x^2} d^4}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {11 b c^3 \left (2 c^2 d^2+e^2\right ) i \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^3}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {11 b c i \sqrt {1-c^2 x^2} d^3}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (4 c^2 h d^2+e (2 e h+81 d i)\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (d (2 e h+9 d i) c^2+18 e^2 i\right ) \sqrt {1-c^2 x^2} d^2}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c (2 e h+27 d i) \sqrt {1-c^2 x^2} d^2}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 d^2 g c^4+\left (-18 i d^2-18 e h d+e^2 g\right ) c^2-36 e^2 i\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c \left (4 e^2 (e h+6 d i)-c^2 d \left (6 i d^2-2 e h d+e^2 g\right )\right ) \sqrt {1-c^2 x^2} d}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (-18 i d^2-6 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {i b i \sin ^{-1}(c x)^2}{2 e^4}-\frac {(e h-3 d i) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}+\frac {b c \left (4 d^2 f c^4+\left (6 h d^2-9 e g d+2 e^2 f\right ) c^2+12 e^2 h\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b i \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b i \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b i \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {i \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {i b i \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b i \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (-2 h d^2-e g d+2 e^2 f\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (6 h d^2-3 e g d+2 e^2 f\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

(b*c*(2*e^2*f - 3*d*e*g + 6*d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^2) - (11*b*c*d^3*i*Sqr
t[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d^2*(2*e*h + 27*d*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c
^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d*(e^2*g - 6*d*e*h - 18*d^2*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d
+ e*x)^2) - (b*c*(2*e^2*(e*g - 4*d*h) - c^2*d*(2*e^2*f - d*e*g - 2*d^2*h))*Sqrt[1 - c^2*x^2])/(4*e^2*(c^2*d^2
- e^2)^2*(d + e*x)) - (11*b*c^3*d^4*i*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) + (b*c*d^2*(18*e
^2*i + c^2*d*(2*e*h + 9*d*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - (b*c*d*(4*e^2*(e*h + 6*
d*i) - c^2*d*(e^2*g - 2*d*e*h + 6*d^2*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - ((I/2)*b*i*
ArcSin[c*x]^2)/e^4 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(3*e^4*(d + e*x)^3) - ((e^2*g -
2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e*h - 3*d*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e
*x)) + (b*c*(4*c^4*d^2*f + 12*e^2*h + c^2*(2*e^2*f - 9*d*e*g + 6*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 -
e^2]*Sqrt[1 - c^2*x^2])])/(12*e*(c^2*d^2 - e^2)^(5/2)) - (11*b*c^3*d^3*(2*c^2*d^2 + e^2)*i*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^4*(c^2*d^2 - e^2)^(5/2)) + (b*c^3*d^2*(4*c^2*d^2*h + e*(2*e*
h + 81*d*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^3*(c^2*d^2 - e^2)^(5/2)) + (
b*c*d*(2*c^4*d^2*g - 36*e^2*i + c^2*(e^2*g - 18*d*e*h - 18*d^2*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*S
qrt[1 - c^2*x^2])])/(12*e^2*(c^2*d^2 - e^2)^(5/2)) + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - S
qrt[c^2*d^2 - e^2])])/e^4 + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4
- (b*i*ArcSin[c*x]*Log[d + e*x])/e^4 + (i*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I
*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d
^2 - e^2])])/e^4

Rule 12

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

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 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 745

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

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 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 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2404

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cos[x])/
(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (f+g x+h x^2+112 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^4} \, dx &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac {1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^4}\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {1232 d^3}{(d+e x)^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 e (h-1512 x)}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {d e^2 \left (-g-6 h x+2016 x^2\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (2 f+3 g x+6 h x^2\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {672 \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{6 e^4}\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(112 b c) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}-\frac {\left (616 b c d^3\right ) \int \frac {1}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{3 e^4}+\frac {\left (b c d^2\right ) \int \frac {h-1512 x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{3 e^3}-\frac {(b c d) \int \frac {-g-6 h x+2016 x^2}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^2}+\frac {(b c) \int \frac {2 f+3 g x+6 h x^2}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}+\frac {\left (308 b c^3 d^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )}+\frac {\left (b c d^2\right ) \int \frac {2 \left (1512 e+c^2 d h\right )-c^2 (1512 d+e h) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )}-\frac {(b c d) \int \frac {2 \left (\frac {1}{2} d \left (4032-2 c^2 g\right )+6 e h\right )-\left (4032 e-c^2 \left (\frac {2016 d^2}{e}+e g-6 d h\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )}+\frac {(b c) \int \frac {2 \left (2 c^2 d f-3 e g+6 d h\right )-\left (12 e h+c^2 \left (2 e f-3 d g-\frac {6 d^2 h}{e}\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \operatorname {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (308 b c^3 d^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {(112 b c) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (308 b c^3 d^3 \left (2 c^2 d^2+e^2\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 )}{3 e^4 \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\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 )}{6 e^3 \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 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 )}{12 e \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e 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 )}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 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 )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e 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 )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(112 b) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}-\frac {(112 b) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 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 )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e 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 )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 i b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}+\frac {(112 i b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 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 )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e 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 )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {112 i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}\\ \end {align*}

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Mathematica [C]  time = 6.97, size = 1921, normalized size = 1.50 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

(-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(3*e^4*(d + e*x)^3) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(2*e
^4*(d + e*x)^2) + (-(a*e*h) + 3*a*d*i)/(e^4*(d + e*x)) + b*f*(-1/9*(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)
]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[3, 1/2, 1/2, 4, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d
- Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)^2*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(3*e*(d + e*x)^3)) + (a*i*Lo
g[d + e*x])/e^4 + b*h*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x
^2])])/Sqrt[c^2*d^2 - e^2])/e^3 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d +
e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*
x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/e^2 + (d^2*((Sqrt[1 - c^2*x^2]*(-(c*
e^2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^
2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[
e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])
))/(6*e^2)) + b*g*(((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d
*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d +
e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/(2*e) - (d*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d +
3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d
+ e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[
-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e)) + b*i*((-
3*d*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2
- e^2]))/e^4 + (3*d^2*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c
^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*
(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^3) - (d^3*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3
*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^
2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*
x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e^3)
+ (((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e
+ (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*Ar
cSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e
^2])])/e)/e^3)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a i x^{3} + a h x^{2} + a g x + a f + {\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arcsin \left (c x\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((a*i*x^3 + a*h*x^2 + a*g*x + a*f + (b*i*x^3 + b*h*x^2 + b*g*x + b*f)*arcsin(c*x))/(e^4*x^4 + 4*d*e^3*
x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 3.74, size = 5670, normalized size = 4.44 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, a i {\left (\frac {18 \, d e^{2} x^{2} + 27 \, d^{2} e x + 11 \, d^{3}}{e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}} + \frac {6 \, \log \left (e x + d\right )}{e^{4}}\right )} - \frac {{\left (3 \, e x + d\right )} a g}{6 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {{\left (3 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} a h}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {a f}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \int \frac {{\left (b i x^{3} + b h x^{2} + b g x + b f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*a*i*((18*d*e^2*x^2 + 27*d^2*e*x + 11*d^3)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + 6*log(e*x + d)
/e^4) - 1/6*(3*e*x + d)*a*g/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/3*(3*e^2*x^2 + 3*d*e*x + d^2)*
a*h/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3) - 1/3*a*f/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) +
integrate((b*i*x^3 + b*h*x^2 + b*g*x + b*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^4*x^4 + 4*d*e^3*x^3
+ 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^4,x)

[Out]

int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**4,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x + h*x**2 + i*x**3)/(d + e*x)**4, x)

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