Integrand size = 31, antiderivative size = 1281 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]
-5/96*b*c^3*d^2*f^3*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/21*b*c*d ^2*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-19/441*b*c^3*d^2*g^3*x^ 7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/81*b*c^5*d^2*g^3*x^9*(-c^2*d*x ^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/63*b*d^2*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/( -c^2*x^2+1)^(1/2)+25/96*b*c*d^2*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^ (1/2)-1/189*b*d^2*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-15/256 *d^2*f*g^2*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/ 2)-3/7*b*d^2*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-15/256*b*d^ 2*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/36*b*d^2*f^3*(-c^2 *x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c-5/32*d^2*f^3*(a+b*arccos(c*x))^2*(-c^ 2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+3/7*b*c*d^2*f^2*g*x^3*(-c^2*d*x^2+ d)^(1/2)/(-c^2*x^2+1)^(1/2)+59/256*b*c*d^2*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)-9/35*b*c^3*d^2*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2 +1)^(1/2)-17/96*b*c^3*d^2*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2 )+3/49*b*c^5*d^2*f^2*g*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+3/64*b* c^5*d^2*f*g^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-15/128*d^2*f*g^2 *x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+5/16*d^2*f*g^2*x^3*(-c^2*x^2 +1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+3/8*d^2*f*g^2*x^3*(-c^2*x^2+1)^ 2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-3/7*d^2*f^2*g*(-c^2*x^2+1)^3*(a+b *arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+15/64*d^2*f*g^2*x^3*(a+b*arccos(...
Time = 9.41 (sec) , antiderivative size = 1582, normalized size of antiderivative = 1.23 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx =\text {Too large to display} \]
Sqrt[-(d*(-1 + c^2*x^2))]*(-1/63*(a*d^2*g*(27*c^2*f^2 + 2*g^2))/c^4 + (a*d ^2*f*(88*c^2*f^2 - 15*g^2)*x)/(128*c^2) - (a*d^2*g*(-81*c^2*f^2 + g^2)*x^2 )/(63*c^2) - (a*d^2*f*(104*c^2*f^2 - 177*g^2)*x^3)/192 + (a*d^2*g*(-27*c^2 *f^2 + 5*g^2)*x^4)/21 + (a*c^2*d^2*f*(8*c^2*f^2 - 51*g^2)*x^5)/48 - (a*c^2 *d^2*g*(-27*c^2*f^2 + 19*g^2)*x^6)/63 + (3*a*c^4*d^2*f*g^2*x^7)/8 + (a*c^4 *d^2*g^3*x^8)/9) - (5*a*d^(5/2)*f*(8*c^2*f^2 + 3*g^2)*ArcTan[(c*x*Sqrt[-(d *(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(128*c^3) - (b*d^2*f^2*g*Sqr t[d*(1 - c^2*x^2)]*(9*c*x + 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] - Cos[3*Arc Cos[c*x]]))/(12*c^2*Sqrt[1 - c^2*x^2]) - (b*d^2*f^2*g*Sqrt[d*(1 - c^2*x^2) ]*(55125*c*x - 1225*Cos[3*ArcCos[c*x]] + 840*(1 - c^2*x^2)^(3/2)*ArcCos[c* x]*(157 + 108*Cos[2*ArcCos[c*x]] + 15*Cos[4*ArcCos[c*x]]) - 1323*Cos[5*Arc Cos[c*x]] - 225*Cos[7*ArcCos[c*x]]))/(235200*c^2*Sqrt[1 - c^2*x^2]) + (b*d ^2*g^3*Sqrt[d*(1 - c^2*x^2)]*(55125*c*x - 1225*Cos[3*ArcCos[c*x]] + 840*(1 - c^2*x^2)^(3/2)*ArcCos[c*x]*(157 + 108*Cos[2*ArcCos[c*x]] + 15*Cos[4*Arc Cos[c*x]]) - 1323*Cos[5*ArcCos[c*x]] - 225*Cos[7*ArcCos[c*x]]))/(352800*c^ 4*Sqrt[1 - c^2*x^2]) + (b*d^2*f^3*Sqrt[d*(1 - c^2*x^2)]*(Cos[2*ArcCos[c*x] ] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])))/(8*c*Sqrt[1 - c^2* x^2]) + (b*d^2*f^3*Sqrt[d*(1 - c^2*x^2)]*(8*ArcCos[c*x]^2 - Cos[4*ArcCos[c *x]] - 4*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/(64*c*Sqrt[1 - c^2*x^2]) - (3*b* d^2*f*g^2*Sqrt[d*(1 - c^2*x^2)]*(8*ArcCos[c*x]^2 - Cos[4*ArcCos[c*x]] -...
Time = 1.38 (sec) , antiderivative size = 630, normalized size of antiderivative = 0.49, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5277, 5263, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^3 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) f^3+3 g x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) f+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {15 f g^2 (a+b \arccos (c x))^2}{256 b c^3}+\frac {1}{6} f^3 x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{24} f^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {5}{16} f^3 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3 f^2 g \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}-\frac {15 f g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{128 c^2}+\frac {3}{8} f g^2 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{16} f g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {15}{64} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {g^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arccos (c x))}{9 c^4}-\frac {g^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {5 f^3 (a+b \arccos (c x))^2}{32 b c}+\frac {3}{49} b c^5 f^2 g x^7+\frac {3}{64} b c^5 f g^2 x^8+\frac {1}{81} b c^5 g^3 x^9-\frac {5}{96} b c^3 f^3 x^4-\frac {9}{35} b c^3 f^2 g x^5-\frac {17}{96} b c^3 f g^2 x^6-\frac {19}{441} b c^3 g^3 x^7-\frac {2 b g^3 x}{63 c^3}-\frac {b f^3 \left (1-c^2 x^2\right )^3}{36 c}+\frac {25}{96} b c f^3 x^2+\frac {3}{7} b c f^2 g x^3-\frac {3 b f^2 g x}{7 c}+\frac {59}{256} b c f g^2 x^4-\frac {15 b f g^2 x^2}{256 c}+\frac {1}{21} b c g^3 x^5-\frac {b g^3 x^3}{189 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((-3*b*f^2*g*x)/(7*c) - (2*b*g^3*x)/(63*c^3) + (2 5*b*c*f^3*x^2)/96 - (15*b*f*g^2*x^2)/(256*c) + (3*b*c*f^2*g*x^3)/7 - (b*g^ 3*x^3)/(189*c) - (5*b*c^3*f^3*x^4)/96 + (59*b*c*f*g^2*x^4)/256 - (9*b*c^3* f^2*g*x^5)/35 + (b*c*g^3*x^5)/21 - (17*b*c^3*f*g^2*x^6)/96 + (3*b*c^5*f^2* g*x^7)/49 - (19*b*c^3*g^3*x^7)/441 + (3*b*c^5*f*g^2*x^8)/64 + (b*c^5*g^3*x ^9)/81 - (b*f^3*(1 - c^2*x^2)^3)/(36*c) + (5*f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/16 - (15*f*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(1 28*c^2) + (15*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/64 + (5*f^3 *x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/24 + (5*f*g^2*x^3*(1 - c^2*x^2 )^(3/2)*(a + b*ArcCos[c*x]))/16 + (f^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos [c*x]))/6 + (3*f*g^2*x^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/8 - (3*f ^2*g*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/(7*c^2) - (g^3*(1 - c^2*x^2) ^(7/2)*(a + b*ArcCos[c*x]))/(7*c^4) + (g^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcC os[c*x]))/(9*c^4) - (5*f^3*(a + b*ArcCos[c*x])^2)/(32*b*c) - (15*f*g^2*(a + b*ArcCos[c*x])^2)/(256*b*c^3)))/Sqrt[1 - c^2*x^2]
3.1.10.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[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_.) + ArcCos[(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*ArcCos[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 = 1.74 (sec) , antiderivative size = 3019, normalized size of antiderivative = 2.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3019\) |
| parts | \(\text {Expression too large to display}\) | \(3019\) |
a*(f^3*(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^3*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4 *(-c^2*d*x^2+d)^(7/2))+3*f*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))))))-3/7*f^2*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)*arccos(c*x)^2*f*(8*c^2*f^2 +3*g^2)*d^2+1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*I*(-c^2*x^2+1)^(1/2)*x^9*c ^9+256*c^10*x^10-576*I*(-c^2*x^2+1)^(1/2)*x^7*c^7-704*c^8*x^8+432*I*(-c^2* x^2+1)^(1/2)*x^5*c^5+688*c^6*x^6-120*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-280*c^4* x^4+9*I*(-c^2*x^2+1)^(1/2)*x*c+41*c^2*x^2-1)*g^3*(I+9*arccos(c*x))*d^2/c^4 /(c^2*x^2-1)+3/16384*(-d*(c^2*x^2-1))^(1/2)*(128*I*(-c^2*x^2+1)^(1/2)*x^8* c^8+128*c^9*x^9-256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x ^2+1)^(1/2)*x^4*c^4+272*c^5*x^5-32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3 +I*(-c^2*x^2+1)^(1/2)+8*c*x)*f*g^2*(8*arccos(c*x)+I)*d^2/c^3/(c^2*x^2-1)+3 /25088*(-d*(c^2*x^2-1))^(1/2)*(64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+64*c^8*x^8- 112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c ^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*g*(28*arccos(c*x)* c^2*f^2+4*I*c^2*f^2-7*arccos(c*x)*g^2-I*g^2)*d^2/c^4/(c^2*x^2-1)+1/2304...
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a* d^2*f^3 + (3*a*c^4*d^2*f^2*g - 2*a*c^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 - 6*a *c^2*d^2*f*g^2)*x^4 - (6*a*c^2*d^2*f^2*g - a*d^2*g^3)*x^3 - (2*a*c^2*d^2*f ^3 - 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b *d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b* c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d ), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
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^3 + 1/128*(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*f*g^2 - 1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(- c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*g^3 - 3/7*(-c^2*d*x^2 + d)^(7/2)*a*f^2*g/( c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3* b*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b *c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^ 3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*ar ctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]