Integrand size = 26, antiderivative size = 488 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b h \arcsin (c x)^2}{2 e^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}-\frac {b c \left (2 e^2 (e g-2 d h)-c^2 d \left (e^2 f+d e g-3 d^2 h\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b h \arcsin (c x) \log (d+e x)}{e^3}+\frac {h (a+b \arcsin (c x)) \log (d+e x)}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3} \]
-1/2*I*b*h*arcsin(c*x)^2/e^3-1/2*(d^2*h-d*e*g+e^2*f)*(a+b*arcsin(c*x))/e^3 /(e*x+d)^2-(-2*d*h+e*g)*(a+b*arcsin(c*x))/e^3/(e*x+d)-1/2*b*c*(2*e^2*(-2*d *h+e*g)-c^2*d*(-3*d^2*h+d*e*g+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^3/(c^2*d^2-e^2)^(3/2)-b*h*arcsin(c*x)*ln(e*x+d)/e ^3+h*(a+b*arcsin(c*x))*ln(e*x+d)/e^3+b*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^3+b*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^3-I*b*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^3-I*b*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^3+1/2*b*c*(d^ 2*h-d*e*g+e^2*f)*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)/(e*x+d)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 7.06 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.04 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {-a e^2 f+a d e g-a d^2 h}{2 e^3 (d+e x)^2}+\frac {-a e g+2 a d h}{e^3 (d+e x)}+b f \left (-\frac {c \sqrt {1+\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}} \sqrt {1+\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x}} \operatorname {AppellF1}\left (2,\frac {1}{2},\frac {1}{2},3,-\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x},-\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}\right )}{4 e^2 (d+e x) \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{2 e (d+e x)^2}\right )+\frac {a h \log (d+e x)}{e^3}+b g \left (\frac {-\frac {\arcsin (c x)}{d+e x}+\frac {c \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}}}{e^2}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e}\right )+b h \left (-\frac {2 d \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \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}}\right )}{e^3}+\frac {d^2 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^2}+\frac {-\frac {i \arcsin (c x)^2}{2 e}+\frac {\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}+\frac {\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}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}}{e^2}\right ) \]
(-(a*e^2*f) + a*d*e*g - a*d^2*h)/(2*e^3*(d + e*x)^2) + (-(a*e*g) + 2*a*d*h )/(e^3*(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]) - ArcSin[c*x]/(2*e*(d + e*x)^2)) + (a*h*Log[d + e*x])/e^3 + b*g*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqr t[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^2 - (d*((c*Sq rt[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*h*((-2*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^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^2) + ( ((-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*ArcSin[c *x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcS...
Time = 1.50 (sec) , antiderivative size = 478, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.
| \(\displaystyle \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int \frac {3 h d^2-e (g-4 h x) d-e^2 (f+2 g x)+2 h (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {3 h d^2-e (g-4 h x) d-e^2 (f+2 g x)+2 h (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b c \int \left (\frac {3 h d^2-e g d-e^2 f-2 e (e g-2 d h) x}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {2 h \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{2 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \arcsin (c x))}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \arcsin (c x))}{e^3}-\frac {b c \left (\frac {2 i h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {2 i h \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {2 h \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{c}+\frac {2 h \arcsin (c x) \log (d+e x)}{c}+\frac {i h \arcsin (c x)^2}{c}+\frac {\arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h+d e g+e^2 f\right )\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e^3}\) |
-1/2*((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)^2) - ((e *g - 2*d*h)*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)) + (h*(a + b*ArcSin[c*x])* Log[d + e*x])/e^3 - (b*c*(-((e*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/ ((c^2*d^2 - e^2)*(d + e*x))) + (I*h*ArcSin[c*x]^2)/c + ((2*e^2*(e*g - 2*d* h) - c^2*d*(e^2*f + d*e*g - 3*d^2*h))*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*h*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/c - (2*h*ArcSin[c* x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c + (2*h* ArcSin[c*x]*Log[d + e*x])/c + ((2*I)*h*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/ (c*d - Sqrt[c^2*d^2 - e^2])])/c + ((2*I)*h*PolyLog[2, (I*e*E^(I*ArcSin[c*x ]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c))/(2*e^3)
3.2.2.3.1 Defintions of rubi rules used
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. 2026 vs. \(2 (489 ) = 978\).
Time = 7.00 (sec) , antiderivative size = 2027, normalized size of antiderivative = 4.15
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2027\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2038\) |
| default | \(\text {Expression too large to display}\) | \(2038\) |
a*(-(-2*d*h+e*g)/e^3/(e*x+d)-1/2*(d^2*h-d*e*g+e^2*f)/e^3/(e*x+d)^2+h/e^3*l n(e*x+d))+b/c*(-I/e^3/(c^2*d^2-e^2)*c^3*h*d^2*arcsin(c*x)^2+1/2*c^2*(4*arc sin(c*x)*c^3*d^3*e*h*x-2*arcsin(c*x)*c^3*d^2*e^2*g*x+I*c^3*d*e^3*g*x^2+(-c ^2*x^2+1)^(1/2)*c^2*d^2*e^2*h*x-(-c^2*x^2+1)^(1/2)*c^2*d*e^3*g*x-2*I*c^3*d ^3*e*h*x+2*I*c^3*d^2*e^2*g*x-2*I*c^3*d*e^3*f*x-I*c^3*d^2*e^2*h*x^2-4*arcsi n(c*x)*d*e^3*h*c*x-e^2*c^3*d^2*f*arcsin(c*x)-e*c^3*d^3*g*arcsin(c*x)+e^3*c *g*arcsin(c*x)*d-3*e^2*c*d^2*h*arcsin(c*x)-I*c^3*d^2*e^2*f+I*c^3*d^3*e*g+( -c^2*x^2+1)^(1/2)*c^2*d^3*e*h-(-c^2*x^2+1)^(1/2)*c^2*d^2*e^2*g+(-c^2*x^2+1 )^(1/2)*c^2*d*e^3*f+2*arcsin(c*x)*e^4*g*c*x+e^4*c*f*arcsin(c*x)+3*c^3*d^4* h*arcsin(c*x)-I*c^3*d^4*h-I*c^3*e^4*f*x^2+(-c^2*x^2+1)^(1/2)*c^2*e^4*f*x)/ (c*e*x+c*d)^2/(c^2*d^2-e^2)/e^3+I/e/(c^2*d^2-e^2)*c*h*arcsin(c*x)^2+2*I/e/ (c^2*d^2-e^2)^2*c^3*h*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/(c^2*d^2-e^2)^(3/2)*c^2*g* arctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2))*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))-3 /e^3/(c^2*d^2-e^2)^(3/2)*c^4*d^3*h*arctan(1/2*(2*(I*c*x+(-c^2*x^2+1)^(1/2) )*e+2*I*c*d)/(c^2*d^2-e^2)^(1/2))+e/(c^2*d^2-e^2)^2*c*h*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)))+e/(c^2*d^2-e^2)^2*c*h*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)))-I/e^3/(c^2*d ^2-e^2)^2*c^5*h*d^4*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2...
\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
integral((a*h*x^2 + a*g*x + a*f + (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)
\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,d x \]