Integrand size = 31, antiderivative size = 617 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \arcsin (c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \] Output:
b*(-2*d*i+e*h)*(-c^2*x^2+1)^(1/2)/c/e^3+1/4*b*i*x*(-c^2*x^2+1)^(1/2)/c/e^2 -1/4*b*i*arcsin(c*x)/c^2/e^2-1/2*I*b*(3*d^2*i-2*d*e*h+e^2*g)*arcsin(c*x)^2 /e^4+(-2*d*i+e*h)*x*(a+b*arcsin(c*x))/e^3+1/2*i*x^2*(a+b*arcsin(c*x))/e^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)+b*c*(-d^3*i+d ^2*e*h-d*e^2*g+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)^(1/2)+b*(3*d^2*i-2*d*e*h+e^2*g)*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^2*i-2 *d*e*h+e^2*g)*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^2*i-2*d*e*h+e^2*g)*arcsin(c*x)*ln(e*x+d)/e^4+(3* d^2*i-2*d*e*h+e^2*g)*(a+b*arcsin(c*x))*ln(e*x+d)/e^4-I*b*(3*d^2*i-2*d*e*h+ e^2*g)*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*(3*d^2*i-2*d*e*h+e^2*g)*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
Time = 0.89 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.96 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {\frac {2 b e (e h-2 d i) \sqrt {1-c^2 x^2}}{c}+\frac {b e^2 i x \sqrt {1-c^2 x^2}}{2 c}-\frac {b e^2 i \arcsin (c x)}{2 c^2}-i b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x)^2+2 e (e h-2 d i) x (a+b \arcsin (c x))+e^2 i x^2 (a+b \arcsin (c x))-\frac {2 \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{d+e x}+\frac {2 b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-2 b \left (e^2 g-2 d e h+3 d^2 i\right ) \arcsin (c x) \log (d+e x)+2 \left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x)) \log (d+e x)-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-2 i b \left (e^2 g-2 d e h+3 d^2 i\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{2 e^4} \] Input:
Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]
Output:
((2*b*e*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/c + (b*e^2*i*x*Sqrt[1 - c^2*x^2]) /(2*c) - (b*e^2*i*ArcSin[c*x])/(2*c^2) - I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*A rcSin[c*x]^2 + 2*e*(e*h - 2*d*i)*x*(a + b*ArcSin[c*x]) + e^2*i*x^2*(a + b* ArcSin[c*x]) - (2*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])) /(d + e*x) + (2*b*c*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*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] + 2*b*(e^ 2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c* d) + Sqrt[c^2*d^2 - e^2])] + 2*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*L og[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - 2*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x] + 2*(e^2*g - 2*d*e*h + 3*d^2* i)*(a + b*ArcSin[c*x])*Log[d + e*x] - (2*I)*b*(e^2*g - 2*d*e*h + 3*d^2*i)* PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - (2*I)*b* (e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt [c^2*d^2 - e^2])])/(2*e^4)
Time = 1.96 (sec) , antiderivative size = 613, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int \frac {e^2 i x^2+2 e (e h-2 d i) x+2 \left (3 i d^2-2 e h d+e^2 g\right ) \log (d+e x)-\frac {2 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right )}{d+e x}}{2 e^4 \sqrt {1-c^2 x^2}}dx+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {e^2 i x^2+2 e (e h-2 d i) x+2 \left (3 i d^2-2 e h d+e^2 g\right ) \log (d+e x)-\frac {2 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right )}{d+e x}}{\sqrt {1-c^2 x^2}}dx}{2 e^4}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b c \int \left (\frac {e^2 i x^2}{\sqrt {1-c^2 x^2}}+\frac {2 e (e h-2 d i) x}{\sqrt {1-c^2 x^2}}+\frac {2 \left (3 i d^2-2 e h d+e^2 g\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}-\frac {2 \left (-i d^3+e h d^2-e^2 g d+e^3 f\right )}{(d+e x) \sqrt {1-c^2 x^2}}\right )dx}{2 e^4}+\frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) (a+b \arcsin (c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \arcsin (c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \arcsin (c x))}{e^3}+\frac {i x^2 (a+b \arcsin (c x))}{2 e^2}-\frac {b c \left (\frac {e^2 i \arcsin (c x)}{2 c^3}+\frac {2 i \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 i \left (3 d^2 i-2 d e h+e^2 g\right ) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 \arcsin (c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{c}+\frac {i \arcsin (c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{c}+\frac {2 \arcsin (c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{c}-\frac {2 \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{\sqrt {c^2 d^2-e^2}}-\frac {2 e \sqrt {1-c^2 x^2} (e h-2 d i)}{c^2}-\frac {e^2 i x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 e^4}\) |
Input:
Int[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]
Output:
((e*h - 2*d*i)*x*(a + b*ArcSin[c*x]))/e^3 + (i*x^2*(a + b*ArcSin[c*x]))/(2 *e^2) - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x)) + ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e ^4 - (b*c*((-2*e*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/c^2 - (e^2*i*x*Sqrt[1 - c^2*x^2])/(2*c^2) + (e^2*i*ArcSin[c*x])/(2*c^3) + (I*(e^2*g - 2*d*e*h + 3* d^2*i)*ArcSin[c*x]^2)/c - (2*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*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] - (2*(e^2*g - 2*d*e*h + 3*d^2*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^2*g - 2*d*e*h + 3*d^2*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^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x])/c + ((2*I)*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/c + ((2*I)*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*Arc Sin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c))/(2*e^4)
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. 2872 vs. \(2 (616 ) = 1232\).
Time = 3.84 (sec) , antiderivative size = 2873, normalized size of antiderivative = 4.66
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2873\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2934\) |
| default | \(\text {Expression too large to display}\) | \(2934\) |
Input:
int((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x,method=_RETURNVERBOS E)
Output:
3*b*c^2/e^4*d^4*i*arcsin(c*x)/(c^2*d^2-e^2)*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)))+3*b*c^2/e^4*d ^4*i*arcsin(c*x)/(c^2*d^2-e^2)*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)))+b*c^2/e^2*g*arcsin(c*x)/(c ^2*d^2-e^2)*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)))*d^2+b*c^2/e^2*g*arcsin(c*x)/(c^2*d^2-e^2)*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)))*d^2-2*b*c^2/e^3*h*d^3*arcsin(c*x)/(c^2*d^2-e^2)*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)))-2*b*c^2/e^3*h*d^3*arcsin(c*x)/(c^2*d^2-e^2)*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)))-3*I*b*c^ 2/e^4*d^4*i/(c^2*d^2-e^2)*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)))-3*I*b*c^2/e^4*d^4*i/(c^2*d^2 -e^2)*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)))-I*b*c^2/e^2*g/(c^2*d^2-e^2)*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)))* d^2+2*I*b*c^2/e^3*h*d^3/(c^2*d^2-e^2)*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)))+2*I*b*c^2/e^3*h* d^3/(c^2*d^2-e^2)*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)))-I*b*c^2/e^2*g/(c^2*d^2-e^2)*dilog...
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,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^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**2,x)
Output:
Integral((a + b*asin(c*x))*(f + g*x + h*x**2 + i*x**3)/(d + e*x)**2, x)
Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,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. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,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. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^2,x)
Output:
int(((a + b*asin(c*x))*(f + g*x + h*x^2 + i*x^3))/(d + e*x)^2, x)
\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^2} \, dx=\frac {2 \mathit {asin} \left (c x \right ) b c \,d^{2} e^{2} h x -2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{3} e^{3} h x +2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c d \,e^{5} f x +2 \sqrt {-c^{2} x^{2}+1}\, b \,d^{2} e^{2} h +6 \,\mathrm {log}\left (e x +d \right ) a c \,d^{4} i +2 a c \,e^{4} f x +2 \mathit {asin} \left (c x \right ) b c d \,e^{3} h \,x^{2}+6 \,\mathrm {log}\left (e x +d \right ) a c \,d^{3} e i x -4 \,\mathrm {log}\left (e x +d \right ) a c \,d^{2} e^{2} h x +2 \,\mathrm {log}\left (e x +d \right ) a c d \,e^{3} g x +2 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{3}}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c d \,e^{5} i x -4 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{2} e^{4} h x +2 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c d \,e^{5} g x +a c d \,e^{3} i \,x^{3}+2 \sqrt {-c^{2} x^{2}+1}\, b d \,e^{3} h x -4 \,\mathrm {log}\left (e x +d \right ) a c \,d^{3} e h +2 \,\mathrm {log}\left (e x +d \right ) a c \,d^{2} e^{2} g -6 a c \,d^{3} e i x +4 a c \,d^{2} e^{2} h x -3 a c \,d^{2} e^{2} i \,x^{2}-2 a c d \,e^{3} g x +2 a c d \,e^{3} h \,x^{2}-2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{4} e^{2} h +2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{2} e^{4} f +2 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{3}}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{2} e^{4} i -4 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{3} e^{3} h +2 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b c \,d^{2} e^{4} g}{2 c d \,e^{4} \left (e x +d \right )} \] Input:
int((i*x^3+h*x^2+g*x+f)*(a+b*asin(c*x))/(e*x+d)^2,x)
Output:
(2*asin(c*x)*b*c*d**2*e**2*h*x + 2*asin(c*x)*b*c*d*e**3*h*x**2 + 2*sqrt( - c**2*x**2 + 1)*b*d**2*e**2*h + 2*sqrt( - c**2*x**2 + 1)*b*d*e**3*h*x - 2* int(asin(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*c*d**4*e**2*h - 2*int(asin (c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*c*d**3*e**3*h*x + 2*int(asin(c*x)/ (d**2 + 2*d*e*x + e**2*x**2),x)*b*c*d**2*e**4*f + 2*int(asin(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*c*d*e**5*f*x + 2*int((asin(c*x)*x**3)/(d**2 + 2* d*e*x + e**2*x**2),x)*b*c*d**2*e**4*i + 2*int((asin(c*x)*x**3)/(d**2 + 2*d *e*x + e**2*x**2),x)*b*c*d*e**5*i*x - 4*int((asin(c*x)*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*c*d**3*e**3*h + 2*int((asin(c*x)*x)/(d**2 + 2*d*e*x + e* *2*x**2),x)*b*c*d**2*e**4*g - 4*int((asin(c*x)*x)/(d**2 + 2*d*e*x + e**2*x **2),x)*b*c*d**2*e**4*h*x + 2*int((asin(c*x)*x)/(d**2 + 2*d*e*x + e**2*x** 2),x)*b*c*d*e**5*g*x + 6*log(d + e*x)*a*c*d**4*i - 4*log(d + e*x)*a*c*d**3 *e*h + 6*log(d + e*x)*a*c*d**3*e*i*x + 2*log(d + e*x)*a*c*d**2*e**2*g - 4* log(d + e*x)*a*c*d**2*e**2*h*x + 2*log(d + e*x)*a*c*d*e**3*g*x - 6*a*c*d** 3*e*i*x + 4*a*c*d**2*e**2*h*x - 3*a*c*d**2*e**2*i*x**2 - 2*a*c*d*e**3*g*x + 2*a*c*d*e**3*h*x**2 + a*c*d*e**3*i*x**3 + 2*a*c*e**4*f*x)/(2*c*d*e**4*(d + e*x))