Integrand size = 31, antiderivative size = 1016 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \] Output:
b*i*(-c^2*x^2+1)^(1/2)/c/e^3+5/2*b*c*d^3*i*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2 -e^2)/(e*x+d)-1/2*b*c*d^2*(4*d*i+3*e*h)*(-c^2*x^2+1)^(1/2)/e^3/(c^2*d^2-e^ 2)/(e*x+d)+1/2*b*c*d*(-4*d^2*i+4*d*e*h+e^2*g)*(-c^2*x^2+1)^(1/2)/e^3/(c^2* d^2-e^2)/(e*x+d)+1/2*b*c*(2*d^3*i-2*d*e^2*g+e^3*f)*(-c^2*x^2+1)^(1/2)/e^3/ (c^2*d^2-e^2)/(e*x+d)-I*b*(-3*d*i+e*h)*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*x*(a+b*arcsin(c*x))/e^3-1/2*(-d^3*i +d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))/e^4/(e*x+d)^2-(3*d^2*i-2*d*e*h+e ^2*g)*(a+b*arcsin(c*x))/e^4/(e*x+d)+5/2*b*c^3*d^4*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)^(3/2)-1/2*b*c*d^2*( 3*c^2*d*h+4*e*i)*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)^(3/2)+1/2*b*c*d*(4*e^2*(-2*d*i+e*h)+c^2*(4*d^3*i+d*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^4/(c^2*d ^2-e^2)^(3/2)-1/2*b*c*(2*e^4*g-6*d^2*e^2*i-c^2*(-4*d^4*i+d*e^3*f))*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)^(3/2 )+b*(-3*d*i+e*h)*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*(-3*d*i+e*h)*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*(-3*d*i+e*h)*arcsin(c*x)*ln(e* x+d)/e^4+(-3*d*i+e*h)*(a+b*arcsin(c*x))*ln(e*x+d)/e^4-I*b*(-3*d*i+e*h)*pol ylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4-1/2*I *b*(-3*d*i+e*h)*arcsin(c*x)^2/e^4
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 9.97 (sec) , antiderivative size = 1556, normalized size of antiderivative = 1.53 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^3,x]
Output:
(a*i*x)/e^3 + (-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(2*e^4*(d + e *x)^2) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(e^4*(d + e*x)) + b*f*(-1/4* (c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e )/(d + e*x)]*AppellF1[2, 1/2, 1/2, 3, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)*Sqrt[1 - c^2*x^2]) - A rcSin[c*x]/(2*e*(d + e*x)^2)) + ((a*e*h - 3*a*d*i)*Log[d + e*x])/e^4 + b*g *((-(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^2 - (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])))/(2*e)) + b*i*((Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x])/(c*e^3) + (3*d^2*(-(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 - (d^3*((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)*S qrt[c^2*d^2 - e^2])))/(2*e^3) - (3*d*(((-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 + (A rcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]...
Time = 2.90 (sec) , antiderivative size = 965, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {5252, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right )}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int -\frac {5 i d^3-e (3 h-4 i x) d^2+e^2 (g-4 x (h+i x)) d+e^3 \left (-2 i x^3+2 g x+f\right )-2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {5 i d^3-e (3 h-4 i x) d^2+e^2 (g-4 x (h+i x)) d+e^3 \left (-2 i x^3+2 g x+f\right )-2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b c \int \left (\frac {5 i d^3}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {e (4 i x-3 h) d^2}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {e^2 \left (4 i x^2+4 h x-g\right ) d}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (2 i x^3-2 g x-f\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {2 (e h-3 d i) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{2 e^4}-\frac {(a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4 (d+e x)}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{2 e^4 (d+e x)^2}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i x (a+b \arcsin (c x))}{e^3}+\frac {(e h-3 d i) \log (d+e x) (a+b \arcsin (c x))}{e^4}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}+\frac {b c \left (\frac {5 c^2 i \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^4}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {5 e i \sqrt {1-c^2 x^2} d^3}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {e \left (3 d h c^2+4 e i\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {e (3 e h+4 d i) \sqrt {1-c^2 x^2} d^2}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {\left (\left (4 i d^3+e^2 g d\right ) c^2+4 e^2 (e h-2 d i)\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {e \left (-4 i d^2+4 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i (e h-3 d i) \arcsin (c x)^2}{c}-\frac {\left (2 g e^4-6 d^2 i e^2-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 (e h-3 d i) \arcsin (c x) \log (d+e x)}{c}-\frac {2 i (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 i (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 e i \sqrt {1-c^2 x^2}}{c^2}+\frac {e \left (2 i d^3-2 e^2 g d+e^3 f\right ) \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e^4}\) |
Input:
Int[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^3,x]
Output:
(i*x*(a + b*ArcSin[c*x]))/e^3 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b* ArcSin[c*x]))/(e^4*(d + e*x)) + ((e*h - 3*d*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 + (b*c*((2*e*i*Sqrt[1 - c^2*x^2])/c^2 + (5*d^3*e*i*Sqrt[1 - c^2 *x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (d^2*e*(3*e*h + 4*d*i)*Sqrt[1 - c^2*x ^2])/((c^2*d^2 - e^2)*(d + e*x)) + (d*e*(e^2*g + 4*d*e*h - 4*d^2*i)*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) + (e*(e^3*f - 2*d*e^2*g + 2*d^3*i )*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (I*(e*h - 3*d*i)*ArcSin [c*x]^2)/c + (5*c^2*d^4*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) - (d^2*e*(3*c^2*d*h + 4*e*i)*ArcTan[( e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/ 2) + (d*(4*e^2*(e*h - 2*d*i) + c^2*(d*e^2*g + 4*d^3*i))*ArcTan[(e + c^2*d* x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) - ((2*e ^4*g - 6*d^2*e^2*i - c^2*(d*e^3*f - 4*d^4*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c ^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) + (2*(e*h - 3*d*i )*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] )/c + (2*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c - (2*(e*h - 3*d*i)*ArcSin[c*x]*Log[d + e*x])/c - ((2*I)*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^ 2 - e^2])])/c - ((2*I)*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3616 vs. \(2 (979 ) = 1958\).
Time = 4.13 (sec) , antiderivative size = 3617, normalized size of antiderivative = 3.56
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3617\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3646\) |
| default | \(\text {Expression too large to display}\) | \(3646\) |
Input:
int((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^3,x,method=_RETURNVERBOS E)
Output:
a*(i/e^3*x-1/e^4*(3*d^2*i-2*d*e*h+e^2*g)/(e*x+d)+(-3*d*i+e*h)/e^4*ln(e*x+d )-1/2/e^4*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)/(e*x+d)^2)+b/c*(-3/(c^2*d^2-e^2)^ 2*c*i*d*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^ (1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-1/2*c^2*(-2*arcsin(c*x)*e^5*g*c*x+I*c ^3*d^4*e*h+I*c^3*d^2*e^3*f+(-c^2*x^2+1)^(1/2)*c^2*d^4*e*i-(-c^2*x^2+1)^(1/ 2)*c^2*d^3*e^2*h+(-c^2*x^2+1)^(1/2)*c^2*d^2*e^3*g-(-c^2*x^2+1)^(1/2)*c^2*d *e^4*f-3*e*c^3*d^4*h*arcsin(c*x)-5*e^2*c*d^3*i*arcsin(c*x)+e^3*c^3*d^2*f*a rcsin(c*x)+e^2*c^3*d^3*g*arcsin(c*x)+2*I*c^3*d*e^4*f*x-I*c^3*d^3*e^2*i*x^2 -I*c^3*d*e^4*g*x^2-2*I*c^3*d^4*e*i*x+2*I*c^3*d^3*e^2*h*x-2*I*c^3*d^2*e^3*g *x+I*c^3*d^2*e^3*h*x^2+(-c^2*x^2+1)^(1/2)*c^2*d^3*e^2*i*x-(-c^2*x^2+1)^(1/ 2)*c^2*d^2*e^3*h*x+(-c^2*x^2+1)^(1/2)*c^2*d*e^4*g*x+6*arcsin(c*x)*c^3*d^4* e*i*x-4*arcsin(c*x)*c^3*d^3*e^2*h*x+2*arcsin(c*x)*c^3*d^2*e^3*g*x-I*c^3*d^ 5*i-e^5*c*f*arcsin(c*x)+5*c^3*d^5*i*arcsin(c*x)+3*e^3*c*d^2*h*arcsin(c*x)- e^4*c*g*arcsin(c*x)*d-I*c^3*d^3*e^2*g+I*c^3*e^5*f*x^2-(-c^2*x^2+1)^(1/2)*c ^2*e^5*f*x-6*arcsin(c*x)*d^2*e^3*i*c*x+4*arcsin(c*x)*d*e^4*h*c*x)/(c^2*d^2 -e^2)/(c*e*x+c*d)^2/e^4+3*I/e^4/(c^2*d^2-e^2)^2*c^5*i*d^5*dilog((I*d*c+(I* c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2 )))-6*I/e^2/(c^2*d^2-e^2)^2*c^3*i*d^3*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/ 2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+6/e^2/(c^2*d^2-e ^2)^2*c^3*i*d^3*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^...
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^3,x, algorithm="fr icas")
Output:
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^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate((i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**3,x)
Output:
Integral((a + b*asin(c*x))*(f + g*x + h*x**2 + i*x**3)/(d + e*x)**3, x)
Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^3,x, algorithm="ma xima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^3,x, algorithm="gi ac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^3,x)
Output:
int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^3, x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
int((i*x^3+h*x^2+g*x+f)*(a+b*asin(c*x))/(e*x+d)^3,x)
Output:
(2*asin(c*x)*b*c*d**3*e*i*x + 4*asin(c*x)*b*c*d**2*e**2*i*x**2 + 2*asin(c* x)*b*c*d*e**3*i*x**3 + 2*sqrt( - c**2*x**2 + 1)*b*d**3*e*i + 4*sqrt( - c** 2*x**2 + 1)*b*d**2*e**2*i*x + 2*sqrt( - c**2*x**2 + 1)*b*d*e**3*i*x**2 - 2 *int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**6 *e*i - 4*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)* b*c*d**5*e**2*i*x - 2*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e **3*x**3),x)*b*c*d**4*e**3*i*x**2 + 2*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3 *d*e**2*x**2 + e**3*x**3),x)*b*c*d**3*e**4*f + 4*int(asin(c*x)/(d**3 + 3*d **2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**2*e**5*f*x + 2*int(asin(c*x )/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d*e**6*f*x**2 - 6 *int((asin(c*x)*x**2)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b *c*d**4*e**3*i + 2*int((asin(c*x)*x**2)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**3*e**4*h - 12*int((asin(c*x)*x**2)/(d**3 + 3*d**2* e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**3*e**4*i*x + 4*int((asin(c*x)*x **2)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**2*e**5*h*x - 6*int((asin(c*x)*x**2)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x )*b*c*d**2*e**5*i*x**2 + 2*int((asin(c*x)*x**2)/(d**3 + 3*d**2*e*x + 3*d*e **2*x**2 + e**3*x**3),x)*b*c*d*e**6*h*x**2 - 6*int((asin(c*x)*x)/(d**3 + 3 *d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**5*e**2*i - 12*int((asin(c *x)*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b*c*d**4*e**3...