Integrand size = 31, antiderivative size = 940 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]
1/36*b*d^2*f^2*(-c^2*x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f^2*x*(a +b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-5/128*d^2*g^2*x*(a+b*arcsin(c*x))*(-c ^2*d*x^2+d)^(1/2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1 /2)+5/24*d^2*f^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+5/4 8*d^2*g^2*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/6*d^2* f^2*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/8*d^2*g^2*x^ 3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-2/7*d^2*f*g*(-c^2* x^2+1)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+2/7*b*d^2*f*g*x*(-c^2* d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-25/96*b*c*d^2*f^2*x^2*(-c^2*d*x^2+d)^( 1/2)/(-c^2*x^2+1)^(1/2)+5/256*b*d^2*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x ^2+1)^(1/2)-2/7*b*c*d^2*f*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/ 96*b*c^3*d^2*f^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-59/768*b*c*d^ 2*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+6/35*b*c^3*d^2*f*g*x^5*( -c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+17/288*b*c^3*d^2*g^2*x^6*(-c^2*d*x^ 2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*f*g*x^7*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*g^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1 )^(1/2)+5/32*d^2*f^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^ 2+1)^(1/2)+5/256*d^2*g^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(- c^2*x^2+1)^(1/2)
Time = 0.52 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.41 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (11025 a^2 \left (8 c^2 f^2+g^2\right )+b^2 c^2 x \left (-1960 c^2 f^2 x \left (99-39 c^2 x^2+8 c^4 x^4\right )-4608 f g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )-245 g^2 x \left (-45+177 c^2 x^2-136 c^4 x^4+36 c^6 x^6\right )\right )+210 a b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )+210 b \left (105 a \left (8 c^2 f^2+g^2\right )+b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )\right ) \arcsin (c x)+11025 b^2 \left (8 c^2 f^2+g^2\right ) \arcsin (c x)^2\right )}{564480 b c^3 \sqrt {1-c^2 x^2}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(11025*a^2*(8*c^2*f^2 + g^2) + b^2*c^2*x*(-1960*c ^2*f^2*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) - 4608*f*g*(-35 + 35*c^2*x^2 - 21*c ^4*x^4 + 5*c^6*x^6) - 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6* x^6)) + 210*a*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2 *x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^ 4 + 48*c^6*x^6)) + 210*b*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2]* (768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7 *g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x] + 1102 5*b^2*(8*c^2*f^2 + g^2)*ArcSin[c*x]^2))/(564480*b*c^3*Sqrt[1 - c^2*x^2])
Time = 1.09 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.51, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5276, 5262, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^2 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f^2 (a+b \arcsin (c x)) \left (1-c^2 x^2\right )^{5/2}+g^2 x^2 (a+b \arcsin (c x)) \left (1-c^2 x^2\right )^{5/2}+2 f g x (a+b \arcsin (c x)) \left (1-c^2 x^2\right )^{5/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {5 g^2 (a+b \arcsin (c x))^2}{256 b c^3}+\frac {1}{6} f^2 x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{24} f^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {5}{16} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {2 f g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^2}-\frac {5 g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{8} g^2 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{48} g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {5}{64} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {5 f^2 (a+b \arcsin (c x))^2}{32 b c}-\frac {2}{49} b c^5 f g x^7-\frac {1}{64} b c^5 g^2 x^8+\frac {5}{96} b c^3 f^2 x^4+\frac {6}{35} b c^3 f g x^5+\frac {17}{288} b c^3 g^2 x^6+\frac {b f^2 \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f^2 x^2-\frac {2}{7} b c f g x^3+\frac {2 b f g x}{7 c}-\frac {59}{768} b c g^2 x^4+\frac {5 b g^2 x^2}{256 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((2*b*f*g*x)/(7*c) - (25*b*c*f^2*x^2)/96 + (5*b*g ^2*x^2)/(256*c) - (2*b*c*f*g*x^3)/7 + (5*b*c^3*f^2*x^4)/96 - (59*b*c*g^2*x ^4)/768 + (6*b*c^3*f*g*x^5)/35 + (17*b*c^3*g^2*x^6)/288 - (2*b*c^5*f*g*x^7 )/49 - (b*c^5*g^2*x^8)/64 + (b*f^2*(1 - c^2*x^2)^3)/(36*c) + (5*f^2*x*Sqrt [1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 - (5*g^2*x*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/(128*c^2) + (5*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) )/64 + (5*f^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/24 + (5*g^2*x^3*( 1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/48 + (f^2*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/6 + (g^2*x^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 - (2*f*g*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^2) + (5*f^2*(a + b *ArcSin[c*x])^2)/(32*b*c) + (5*g^2*(a + b*ArcSin[c*x])^2)/(256*b*c^3)))/Sq rt[1 - c^2*x^2]
3.1.41.3.1 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[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
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 = 0.66 (sec) , antiderivative size = 2090, normalized size of antiderivative = 2.22
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2090\) |
| parts | \(\text {Expression too large to display}\) | \(2090\) |
a*(f^2*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d *(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(- c^2*d*x^2+d)^(1/2)))))+g^2*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2*(1/6 *x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c ^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d )^(1/2))))))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+b*(-5/256*(-d*(c^2*x^2-1) )^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*(8*c^2*f^2+g^2)*d ^2+1/3136*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^ 2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56* I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(I+7*arcsin (c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+ 1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18* I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*(6*arc sin(c*x)*c^2*f^2+I*c^2*f^2-6*arcsin(c*x)*g^2-I*g^2)*d^2/c^3/(c^2*x^2-1)-3/ 1024*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(22*I*c^2 *f^2+32*arcsin(c*x)*c^2*f^2+I*g^2+4*arcsin(c*x)*g^2)*cos(3*arcsin(c*x))*d^ 2/c^3/(c^2*x^2-1)-5/64*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2 )*x*c-1)*f*g*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/64*(-d*(c^2*x^2-1))^(1/ 2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(c*x)-I)*d^2/c^2/(c^2*x ^2-1)+1/64*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4...
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
integral((a*c^4*d^2*g^2*x^6 + 2*a*c^4*d^2*f*g*x^5 - 4*a*c^2*d^2*f*g*x^3 + 2*a*d^2*f*g*x + a*d^2*f^2 + (a*c^4*d^2*f^2 - 2*a*c^2*d^2*g^2)*x^4 - (2*a*c ^2*d^2*f^2 - a*d^2*g^2)*x^2 + (b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 - 4 *b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 - 2*b*c^2* d^2*g^2)*x^4 - (2*b*c^2*d^2*f^2 - b*d^2*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d *x^2 + d), x)
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt (-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a*f^2 + 1/384*(8*(-c^2* d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^2*d* x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*ar csin(c*x)/c^3)*a*g^2 - 2/7*(-c^2*d*x^2 + d)^(7/2)*a*f*g/(c^2*d) + sqrt(d)* integrate((b*c^4*d^2*g^2*x^6 + 2*b*c^4*d^2*f*g*x^5 - 4*b*c^2*d^2*f*g*x^3 + 2*b*d^2*f*g*x + b*d^2*f^2 + (b*c^4*d^2*f^2 - 2*b*c^2*d^2*g^2)*x^4 - (2*b* c^2*d^2*f^2 - b*d^2*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sq rt(c*x + 1)*sqrt(-c*x + 1)), x)
Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]