Integrand size = 31, antiderivative size = 1073 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}+\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^3 \sqrt {1-c^2 x^2}}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c g^2 (f+g x)}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}} \]
-a*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/g^3-b*d*(c*f-g)*(c*f+g)*arcsin(c *x)*(-c^2*d*x^2+d)^(1/2)/g^3+1/2*c^2*d*f*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d )^(1/2)/g^2+1/3*d*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g-1/ 3*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+b*c*d*(c*f-g)*(c*f+g)* x*(-c^2*d*x^2+d)^(1/2)/g^3/(-c^2*x^2+1)^(1/2)-1/4*b*c^3*d*f*x^2*(-c^2*d*x^ 2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)+1/9*b*c^3*d*x^3*(-c^2*d*x^2+d)^(1/2)/g/( -c^2*x^2+1)^(1/2)+1/4*c*d*f*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^2 /(-c^2*x^2+1)^(1/2)-1/2*c*d*(c*f-g)*(c*f+g)*x*(a+b*arcsin(c*x))^2*(-c^2*d* x^2+d)^(1/2)/b/g^3/(-c^2*x^2+1)^(1/2)-1/2*d*(c^2*f^2-g^2)^2*(a+b*arcsin(c* x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/g^4/(g*x+f)/(-c^2*x^2+1)^(1/2)+a*d*(c^2*f^2 -g^2)^(3/2)*arctan((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(-c ^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-I*b*d*(c^2*f^2-g^2)^(3/2)*arcsin( c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2* d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)+I*b*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x )*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x ^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*(I* c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/ g^4/(-c^2*x^2+1)^(1/2)+b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*(I*c*x+(-c^2*x^ 2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^ 2+1)^(1/2)-1/2*d*(c*f-g)*(c*f+g)*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)...
Time = 1.19 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-9 b c^3 f x^2+4 b c g x \left (-3+c^2 x^2\right )+18 c^2 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+12 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {9 c f (a+b \arcsin (c x))^2}{b}+\frac {18 \left (c^2 f^2-g^2\right ) \left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2}{b c (f+g x)}-\frac {18 \left (c^2 f^2-g^2\right ) \left (\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2+c^2 g x (f+g x) (a+b \arcsin (c x))^2-2 b c (f+g x) \left (b c g x-g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{b c g^2 (f+g x)}\right )}{36 g^2 \sqrt {1-c^2 x^2}} \]
(d*Sqrt[d - c^2*d*x^2]*(-9*b*c^3*f*x^2 + 4*b*c*g*x*(-3 + c^2*x^2) + 18*c^2 *f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + 12*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (9*c*f*(a + b*ArcSin[c*x])^2)/b + (18*(c^2*f^2 - g^2)*(- 1 + c^2*x^2)*(a + b*ArcSin[c*x])^2)/(b*c*(f + g*x)) - (18*(c^2*f^2 - g^2)* ((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c *x])^2 - 2*b*c*(f + g*x)*(b*c*g*x - g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x] ) - I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x ])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c *f + Sqrt[c^2*f^2 - g^2])]) - I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sq rt[c^2*f^2 - g^2])]))))/(b*c*g^2*(f + g*x))))/(36*g^2*Sqrt[1 - c^2*x^2])
Time = 2.34 (sec) , antiderivative size = 709, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5276, 5266, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x}dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5266 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) c^2}{g}+\frac {f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) c^2}{g^2}+\frac {\left (g^2-c^2 f^2\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^2 (f+g x)}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (-\frac {\left (1-c^2 x^2\right ) \left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2}{2 b c g^2 (f+g x)}-\frac {\left (c^2 f^2-g^2\right )^2 (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {c x \left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2}{2 b g^3}+\frac {c^2 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 g}+\frac {c f (a+b \arcsin (c x))^2}{4 b g^2}+\frac {a \left (c^2 f^2-g^2\right )^{3/2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g^3}-\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {i b \arcsin (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {i b \arcsin (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^4}-\frac {b \sqrt {1-c^2 x^2} \arcsin (c x) (c f-g) (c f+g)}{g^3}-\frac {b c^3 f x^2}{4 g^2}+\frac {b c^3 x^3}{9 g}+\frac {b c x \left (c^2 f^2-g^2\right )}{g^3}-\frac {b c x}{3 g}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*(-1/3*(b*c*x)/g + (b*c*(c^2*f^2 - g^2)*x)/g^3 - (b* c^3*f*x^2)/(4*g^2) + (b*c^3*x^3)/(9*g) - (a*(c*f - g)*(c*f + g)*Sqrt[1 - c ^2*x^2])/g^3 - (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/g^3 + (c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*g^2) + ((1 - c^2*x^2)^ (3/2)*(a + b*ArcSin[c*x]))/(3*g) + (c*f*(a + b*ArcSin[c*x])^2)/(4*b*g^2) - (c*(c^2*f^2 - g^2)*x*(a + b*ArcSin[c*x])^2)/(2*b*g^3) - ((c^2*f^2 - g^2)^ 2*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)) - ((c^2*f^2 - g^2)*(1 - c^2 *x^2)*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)) + (a*(c^2*f^2 - g^2)^(3 /2)*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/g^4 - ( I*b*(c^2*f^2 - g^2)^(3/2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^4 + (I*b*(c^2*f^2 - g^2)^(3/2)*ArcSin[c*x]*Log [1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^4 - (b*(c^2*f ^2 - g^2)^(3/2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g ^2])])/g^4 + (b*(c^2*f^2 - g^2)^(3/2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/( c*f + Sqrt[c^2*f^2 - g^2])])/g^4))/Sqrt[1 - c^2*x^2]
3.1.39.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Time = 0.64 (sec) , antiderivative size = 1546, normalized size of antiderivative = 1.44
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1546\) |
| parts | \(\text {Expression too large to display}\) | \(1546\) |
a/g*(1/3*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)+ c^2*d*f/g*(-1/4*(-2*(x+f/g)*c^2*d+2*c^2*d*f/g)/c^2/d*(-(x+f/g)^2*c^2*d+2*c ^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)-1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g ^2-4*c^4*d^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g) ^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)))-d*(c^2*f^2-g^2)/ g^2*((-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)+c^2* d*f/g/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*( x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+d*(c^2*f^2-g^2)/g^2/(-d*(c^2*f^2-g^2)/g ^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^ 2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^( 1/2))/(x+f/g))))+b*(1/4*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2 -1)*arcsin(c*x)^2*f*(2*c^2*f^2-3*g^2)*d*c/g^4-1/72*(-d*(c^2*x^2-1))^(1/2)* (4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2) *x*c+1)*(I+3*arcsin(c*x))*d/(c^2*x^2-1)/g+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2* I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(I+2* arcsin(c*x))*d*c/(c^2*x^2-1)/g^2-1/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c ^2*x^2+1)^(1/2)*x*c-1)*(4*c^2*f^2-5*g^2)*(arcsin(c*x)+I)*d/(c^2*x^2-1)/g^3 -1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(4*c^2*f^ 2-5*g^2)*(arcsin(c*x)-I)*d/(c^2*x^2-1)/g^3+1/16*(-d*(c^2*x^2-1))^(1/2)*(2* I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-...
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{g x + f} \,d x } \]
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arcsin(c*x))*sqrt(-c^2* d*x^2 + d)/(g*x + f), x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor e details)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]