Integrand size = 31, antiderivative size = 213 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (2 f g+\left (c^2 f^2+g^2\right ) x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^2 \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {d-c^2 d x^2}} \]
(2*f*g+(c^2*f^2+g^2)*x)*(a+b*arcsin(c*x))/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/2*g ^2*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/b/c^3/d/(-c^2*d*x^2+d)^(1/2)+1/2 *b*(c*f+g)^2*ln(-c*x+1)*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+1/2* b*(c*f-g)^2*ln(c*x+1)*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.73 \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\frac {g^2 (a+b \arcsin (c x))^2}{b}+(-c f+g)^2 \left (-\left ((a+b \arcsin (c x)) \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+(c f+g)^2 \left (2 b \log \left (\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+(a+b \arcsin (c x)) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{2 c^3 d \sqrt {d-c^2 d x^2}} \]
(Sqrt[1 - c^2*x^2]*(-((g^2*(a + b*ArcSin[c*x])^2)/b) + (-(c*f) + g)^2*(-(( a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcS in[c*x])/4]]) + (c*f + g)^2*(2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b *ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^3*d*Sqrt[d - c^2*d*x^2])
Time = 0.61 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5276, 5274, 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.
| \(\displaystyle \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5274 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (f^2 c^2+2 f g x c^2+g^2\right ) (a+b \arcsin (c x))}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {g^2 (a+b \arcsin (c x))}{c^2 \sqrt {1-c^2 x^2}}\right )dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {g^2 (a+b \arcsin (c x))^2}{2 b c^3}+\frac {\left (x \left (c^2 f^2+g^2\right )+2 f g\right ) (a+b \arcsin (c x))}{c^2 \sqrt {1-c^2 x^2}}-\frac {2 b f g \text {arctanh}(c x)}{c^2}+\frac {b \left (c^2 f^2+g^2\right ) \log \left (1-c^2 x^2\right )}{2 c^3}\right )}{d \sqrt {d-c^2 d x^2}}\) |
(Sqrt[1 - c^2*x^2]*(((2*f*g + (c^2*f^2 + g^2)*x)*(a + b*ArcSin[c*x]))/(c^2 *Sqrt[1 - c^2*x^2]) - (g^2*(a + b*ArcSin[c*x])^2)/(2*b*c^3) - (2*b*f*g*Arc Tanh[c*x])/c^2 + (b*(c^2*f^2 + g^2)*Log[1 - c^2*x^2])/(2*c^3)))/(d*Sqrt[d - c^2*d*x^2])
3.1.50.3.1 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[(a + b*ArcSin[c*x] )^n/Sqrt[d + e*x^2], (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] && ILtQ[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]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.29
| method | result | size |
| default | \(a \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} f^{2}+g^{2}\right ) \arcsin \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right ) \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}-2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}+2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(488\) |
| parts | \(a \left (\frac {f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+g^{2} \left (\frac {x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}\right )+\frac {2 f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} f^{2}+g^{2}\right ) \arcsin \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right ) \left (c^{2} f^{2}+g^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, c f g +2 x \,c^{2} f g \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}-2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (c^{2} f^{2}+2 c f g +g^{2}\right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(488\) |
a*(f^2/d*x/(-c^2*d*x^2+d)^(1/2)+g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/ (c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+2*f*g/c^2/d/(- c^2*d*x^2+d)^(1/2))+b*(1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c ^3/(c^2*x^2-1)*g^2*arcsin(c*x)^2+2*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^( 1/2)/d^2/c^3/(c^2*x^2-1)*(c^2*f^2+g^2)*arcsin(c*x)-(-d*(c^2*x^2-1))^(1/2)* (c*x+I*(-c^2*x^2+1)^(1/2))*arcsin(c*x)*(c^2*f^2+g^2-2*I*(-c^2*x^2+1)^(1/2) *c*f*g+2*x*c^2*f*g)/d^2/c^3/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1 )^(1/2)/d^2/c^3/(c^2*x^2-1)*(c^2*f^2-2*c*f*g+g^2)*ln(I*c*x+(-c^2*x^2+1)^(1 /2)+I)-(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*(c^2* f^2+2*c*f*g+g^2)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I))
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*arcsin(c*x))/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
a*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^2 *x*arcsin(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a*f^2*x/(sqrt(-c^2*d*x^2 + d)*d) + sqrt(d)*integrate((b*g^2*x^2 + 2*b*f*g*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)* arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x) - 1/2*b*f^2*log(x^2 - 1/c^2)/(c*d^(3/2)) + 2*a*f*g/(sqrt(-c^2*d*x ^2 + d)*c^2*d)
Exception generated. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {(f+g x)^2 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]