Integrand size = 35, antiderivative size = 920 \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=-\frac {4 b^2 h^2 x}{9 c^2}-\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) x}{e^2}-\frac {b^2 d h^2 x^2}{2 e}-\frac {2}{27} b^2 h^2 x^3+\frac {a b h \left (4 e^2 h+c^2 \left (36 e^2 f-25 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{9 c^3 e^2}+\frac {5 a b d h^2 (d+e x) \sqrt {1-c^2 x^2}}{9 c e^2}+\frac {2 a b h^2 (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c e^2}-\frac {a b d \left (2 c^2 d^2+3 e^2\right ) h^2 \arcsin (c x)}{3 c^2 e^3}+\frac {4 b^2 h^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c^3}+\frac {2 b^2 h \left (2 e^2 f-d^2 h\right ) \sqrt {1-c^2 x^2} \arcsin (c x)}{c e^2}+\frac {b^2 d h^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c e}+\frac {2 b^2 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{9 c}-\frac {b^2 d^3 h^2 \arcsin (c x)^2}{3 e^3}-\frac {b^2 d h^2 \arcsin (c x)^2}{2 c^2 e}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}+\frac {2 a b c \left (e^2 f-d^2 h\right )^2 \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}} \]
-4/9*b^2*h^2*x/c^2-2*b^2*h*(-d^2*h+2*e^2*f)*x/e^2-1/2*b^2*d*h^2*x^2/e-2/27 *b^2*h^2*x^3-1/3*a*b*d*(2*c^2*d^2+3*e^2)*h^2*arcsin(c*x)/c^2/e^3-1/3*b^2*d ^3*h^2*arcsin(c*x)^2/e^3-1/2*b^2*d*h^2*arcsin(c*x)^2/c^2/e+2*h*(-d^2*h+e^2 *f)*x*(a+b*arcsin(c*x))^2/e^2-(-d^2*h+e^2*f)^2*(a+b*arcsin(c*x))^2/e^3/(e* x+d)+1/3*h^2*(e*x+d)^3*(a+b*arcsin(c*x))^2/e^3+2*a*b*c*(-d^2*h+e^2*f)^2*ar ctan((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) ^(1/2)+2*I*b^2*c*(-d^2*h+e^2*f)^2*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/(c^2*d^2-e^2)^(1/2)-2*I*b^2*c*(-d^2 *h+e^2*f)^2*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/(c^2*d^2-e^2)^(1/2)-2*b^2*c*(-d^2*h+e^2*f)^2*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/(c^2*d^2-e^2)^ (1/2)+2*b^2*c*(-d^2*h+e^2*f)^2*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/(c^2*d^2-e^2)^(1/2)+1/9*a*b*h*(4*e^2*h+c^2*(- 25*d^2*h+36*e^2*f))*(-c^2*x^2+1)^(1/2)/c^3/e^2+5/9*a*b*d*h^2*(e*x+d)*(-c^2 *x^2+1)^(1/2)/c/e^2+2/9*a*b*h^2*(e*x+d)^2*(-c^2*x^2+1)^(1/2)/c/e^2+4/9*b^2 *h^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3+2*b^2*h*(-d^2*h+2*e^2*f)*arcsin(c* x)*(-c^2*x^2+1)^(1/2)/c/e^2+b^2*d*h^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/e +2/9*b^2*h^2*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c
Time = 0.95 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.57 \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\frac {h \left (2 e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}+\frac {d h^2 x^2 (a+b \arcsin (c x))^2}{e}+\frac {1}{3} h^2 x^3 (a+b \arcsin (c x))^2-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}-\frac {2 b h^2 \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )-3 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^3}-\frac {2 b h \left (2 e^2 f-d^2 h\right ) \left (b x-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}\right )}{e^2}-\frac {b d h^2 \left (b x^2-\frac {2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {(a+b \arcsin (c x))^2}{b c^2}\right )}{2 e}+\frac {2 b c \left (e^2 f-d^2 h\right )^2 \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e^3 \sqrt {c^2 d^2-e^2}} \]
(h*(2*e^2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 + (d*h^2*x^2*(a + b*ArcS in[c*x])^2)/e + (h^2*x^3*(a + b*ArcSin[c*x])^2)/3 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) - (2*b*h^2*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*Arc Sin[c*x]))/(27*c^3) - (2*b*h*(2*e^2*f - d^2*h)*(b*x - (Sqrt[1 - c^2*x^2]*( a + b*ArcSin[c*x]))/c))/e^2 - (b*d*h^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/(2*e) + (2*b*c*(e^2* f - d^2*h)^2*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(- (c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqr t[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2 *d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(e^3*Sqrt[c^2*d^2 - e^2])
Time = 4.01 (sec) , antiderivative size = 888, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {5256, 27, 5298, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2 \left (2 d h x+e f+e h x^2\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 5256 |
| \(\displaystyle -2 b c \int \frac {\left (h^2 (d+e x)^3+6 e h \left (e^2 f-d^2 h\right ) x-\frac {3 \left (e^2 f-d^2 h\right )^2}{d+e x}\right ) (a+b \arcsin (c x))}{3 e^3 \sqrt {1-c^2 x^2}}dx+\frac {2 h x \left (e^2 f-d^2 h\right ) (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b c \int \frac {\left (h^2 (d+e x)^3+6 e h \left (e^2 f-d^2 h\right ) x-\frac {3 \left (e^2 f-d^2 h\right )^2}{d+e x}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 e^3}+\frac {2 h x \left (e^2 f-d^2 h\right ) (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}\) |
| \(\Big \downarrow \) 5298 |
| \(\displaystyle -\frac {2 b c \int \left (\frac {b \arcsin (c x) \left (-2 h^2 d^4+6 e^2 f h d^2+4 e^3 h^2 x^3 d+2 e h \left (3 e^2 f-d^2 h\right ) x d+e^4 h^2 x^4-3 e^4 f^2+6 e^4 f h x^2\right )}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {a \left (-2 h^2 d^4+6 e^2 f h d^2+4 e^3 h^2 x^3 d+2 e h \left (3 e^2 f-d^2 h\right ) x d+e^4 h^2 x^4-3 e^4 f^2+6 e^4 f h x^2\right )}{(d+e x) \sqrt {1-c^2 x^2}}\right )dx}{3 e^3}+\frac {2 h x \left (e^2 f-d^2 h\right ) (a+b \arcsin (c x))^2}{e^2}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}+\frac {h^2 (d+e x)^3 (a+b \arcsin (c x))^2}{3 e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {h^2 (a+b \arcsin (c x))^2 (d+e x)^3}{3 e^3}+\frac {2 h \left (e^2 f-d^2 h\right ) x (a+b \arcsin (c x))^2}{e^2}-\frac {2 b c \left (\frac {b h^2 \arcsin (c x)^2 d^3}{2 c}+\frac {3 b e^2 h^2 x^2 d}{4 c}+\frac {3 b e^2 h^2 \arcsin (c x)^2 d}{4 c^3}+\frac {a \left (2 c^2 d^2+3 e^2\right ) h^2 \arcsin (c x) d}{2 c^3}-\frac {3 b e^2 h^2 x \sqrt {1-c^2 x^2} \arcsin (c x) d}{2 c^2}-\frac {5 a e h^2 (d+e x) \sqrt {1-c^2 x^2} d}{6 c^2}+\frac {b e^3 h^2 x^3}{9 c}+\frac {2 b e^3 h^2 x}{3 c^3}+\frac {3 b e h \left (2 e^2 f-d^2 h\right ) x}{c}-\frac {2 b e^3 h^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^4}-\frac {b e^3 h^2 x^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^2}-\frac {3 b e h \left (2 e^2 f-d^2 h\right ) \sqrt {1-c^2 x^2} \arcsin (c x)}{c^2}-\frac {3 a \left (e^2 f-d^2 h\right )^2 \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+\frac {3 i b \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{\sqrt {c^2 d^2-e^2}}-\frac {3 i b \left (e^2 f-d^2 h\right )^2 \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{\sqrt {c^2 d^2-e^2}}+\frac {3 b \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{\sqrt {c^2 d^2-e^2}}-\frac {3 b \left (e^2 f-d^2 h\right )^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{\sqrt {c^2 d^2-e^2}}-\frac {a e h^2 (d+e x)^2 \sqrt {1-c^2 x^2}}{3 c^2}-\frac {a e h \left (\left (36 e^2 f-25 d^2 h\right ) c^2+4 e^2 h\right ) \sqrt {1-c^2 x^2}}{6 c^4}\right )}{3 e^3}-\frac {\left (e^2 f-d^2 h\right )^2 (a+b \arcsin (c x))^2}{e^3 (d+e x)}\) |
(2*h*(e^2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) + (h^2*(d + e*x)^3*(a + b*ArcSin[c*x]) ^2)/(3*e^3) - (2*b*c*((2*b*e^3*h^2*x)/(3*c^3) + (3*b*e*h*(2*e^2*f - d^2*h) *x)/c + (3*b*d*e^2*h^2*x^2)/(4*c) + (b*e^3*h^2*x^3)/(9*c) - (a*e*h*(4*e^2* h + c^2*(36*e^2*f - 25*d^2*h))*Sqrt[1 - c^2*x^2])/(6*c^4) - (5*a*d*e*h^2*( d + e*x)*Sqrt[1 - c^2*x^2])/(6*c^2) - (a*e*h^2*(d + e*x)^2*Sqrt[1 - c^2*x^ 2])/(3*c^2) + (a*d*(2*c^2*d^2 + 3*e^2)*h^2*ArcSin[c*x])/(2*c^3) - (2*b*e^3 *h^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^4) - (3*b*e*h*(2*e^2*f - d^2*h)*S qrt[1 - c^2*x^2]*ArcSin[c*x])/c^2 - (3*b*d*e^2*h^2*x*Sqrt[1 - c^2*x^2]*Arc Sin[c*x])/(2*c^2) - (b*e^3*h^2*x^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^2) + (b*d^3*h^2*ArcSin[c*x]^2)/(2*c) + (3*b*d*e^2*h^2*ArcSin[c*x]^2)/(4*c^3) - (3*a*(e^2*f - d^2*h)^2*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] + ((3*I)*b*(e^2*f - d^2*h)^2*ArcSin[c*x] *Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/Sqrt[c^2*d^ 2 - e^2] - ((3*I)*b*(e^2*f - d^2*h)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin [c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/Sqrt[c^2*d^2 - e^2] + (3*b*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/ Sqrt[c^2*d^2 - e^2] - (3*b*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c *x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/Sqrt[c^2*d^2 - e^2]))/(3*e^3)
3.2.21.3.1 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_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x _)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcSin[c*x])^n u, x] - Simp[b*c *n Int[SimplifyIntegrand[u*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2] ), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[ p, 0] && EqQ[e*g - 2*d*h, 0]
Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p _), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) ^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2173 vs. \(2 (886 ) = 1772\).
Time = 3.88 (sec) , antiderivative size = 2174, normalized size of antiderivative = 2.36
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2174\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2208\) |
| default | \(\text {Expression too large to display}\) | \(2208\) |
a^2*(h/e^2*(1/3*x^3*e^2*h+x^2*d*e*h-d^2*h*x+2*e^2*f*x)-(d^4*h^2-2*d^2*e^2* f*h+e^4*f^2)/e^3/(e*x+d))+b^2/c*(1/8/c*d*h^2*(2*I*arcsin(c*x)+2*arcsin(c*x )^2-1)/e*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)-1/8*(c*x+I*(-c^2*x^2+1)^ (1/2))*h*(4*c^2*d^2*h-8*c^2*e^2*f-e^2*h)*(arcsin(c*x)^2-2-2*I*arcsin(c*x)) /c^2/e^2+4*I*(-c^2*d^2+e^2)^(1/2)/e/(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)))*f*h* c^2*d^2+1/8*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*d*h^2*(-2*I*arcsin(c* x)+2*arcsin(c*x)^2-1)/c/e-(d^4*h^2-2*d^2*e^2*f*h+e^4*f^2)*arcsin(c*x)^2*c^ 2/e^3/(c*e*x+c*d)-2*(-c^2*d^2+e^2)^(1/2)/e^3/(c^2*d^2-e^2)*c^2*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)))*d^4*h^2+4*(-c^2*d^2+e^2)^(1/2)/e/(c^2*d^2-e^2)*c^2*arc sin(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)))*d^2*f*h-2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)*e *c^2*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)))*f^2+2*(-c^2*d^2+e^2)^(1/2)/e^3/(c^2*d^ 2-e^2)*c^2*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)))*d^4*h^2-4*(-c^2*d^2+e^2)^(1/2)/e/( c^2*d^2-e^2)*c^2*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)))*d^2*f*h+2*(-c^2*d^2+e^2)^(1/ 2)/(c^2*d^2-e^2)*e*c^2*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))...
\[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
integral((a^2*e^2*h^2*x^4 + 4*a^2*d*e*h^2*x^3 + 4*a^2*d*e*f*h*x + a^2*e^2* f^2 + 2*(a^2*e^2*f*h + 2*a^2*d^2*h^2)*x^2 + (b^2*e^2*h^2*x^4 + 4*b^2*d*e*h ^2*x^3 + 4*b^2*d*e*f*h*x + b^2*e^2*f^2 + 2*(b^2*e^2*f*h + 2*b^2*d^2*h^2)*x ^2)*arcsin(c*x)^2 + 2*(a*b*e^2*h^2*x^4 + 4*a*b*d*e*h^2*x^3 + 4*a*b*d*e*f*h *x + a*b*e^2*f^2 + 2*(a*b*e^2*f*h + 2*a*b*d^2*h^2)*x^2)*arcsin(c*x))/(e^2* x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (2 d h x + e f + e h x^{2}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (e f+2 d h x+e h x^2\right )^2 (a+b \arcsin (c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,h\,x^2+2\,d\,h\,x+e\,f\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]