Integrand size = 31, antiderivative size = 871 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {5 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{96 c}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d f^2 x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {5}{48} d g^2 x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{6} f^2 x \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {1}{8} g^2 x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))-\frac {2 f g \left (d-c^2 d x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2 d}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \] Output:
-2/7*b*d^2*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+5/32*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)+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)-5/96*b*d^2*f^2*(-c^2*x^2+1)^(3/2) *(-c^2*d*x^2+d)^(1/2)/c-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*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))-5/128*d^2*g^2 *x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2+5/64*d^2*g^2*x^3*(-c^2*d*x^2 +d)^(1/2)*(a+b*arccos(c*x))+5/24*d*f^2*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos( c*x))+5/48*d*g^2*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))+1/6*f^2*x*(-c^ 2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))+1/8*g^2*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*a rccos(c*x))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)*(a+b*arccos(c*x))/c^2/d-5/32*d^2* f^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)-5/256* d^2*g^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 3.26 (sec) , antiderivative size = 794, normalized size of antiderivative = 0.91 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 \left (-352800 b \left (8 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \arccos (c x)^2-705600 a \sqrt {d} \left (8 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-2822400 b c^2 f g x-5160960 a c f g \sqrt {1-c^2 x^2}+12418560 a c^3 f^2 x \sqrt {1-c^2 x^2}-705600 a c g^2 x \sqrt {1-c^2 x^2}+15482880 a c^3 f g x^2 \sqrt {1-c^2 x^2}-9784320 a c^5 f^2 x^3 \sqrt {1-c^2 x^2}+5550720 a c^3 g^2 x^3 \sqrt {1-c^2 x^2}-15482880 a c^5 f g x^4 \sqrt {1-c^2 x^2}+3010560 a c^7 f^2 x^5 \sqrt {1-c^2 x^2}-6397440 a c^5 g^2 x^5 \sqrt {1-c^2 x^2}+5160960 a c^7 f g x^6 \sqrt {1-c^2 x^2}+2257920 a c^7 g^2 x^7 \sqrt {1-c^2 x^2}+141120 b \left (15 c^2 f^2+g^2\right ) \cos (2 \arccos (c x))+564480 b c f g \cos (3 \arccos (c x))-211680 b c^2 f^2 \cos (4 \arccos (c x))+35280 b g^2 \cos (4 \arccos (c x))-112896 b c f g \cos (5 \arccos (c x))+15680 b c^2 f^2 \cos (6 \arccos (c x))-15680 b g^2 \cos (6 \arccos (c x))+11520 b c f g \cos (7 \arccos (c x))+2205 b g^2 \cos (8 \arccos (c x))\right )+168 b \sqrt {d-c^2 d x^2} \arccos (c x) \left (-58112 c f g \sqrt {1-c^2 x^2}+111872 c^3 f g x^2 \sqrt {1-c^2 x^2}-27648 c f g \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))-3840 c f g \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+25200 c^2 f^2 \sin (2 \arccos (c x))+1680 g^2 \sin (2 \arccos (c x))-8960 c f g \sin (3 \arccos (c x))-5040 c^2 f^2 \sin (4 \arccos (c x))+840 g^2 \sin (4 \arccos (c x))-5376 c f g \sin (5 \arccos (c x))+560 c^2 f^2 \sin (6 \arccos (c x))-560 g^2 \sin (6 \arccos (c x))+105 g^2 \sin (8 \arccos (c x))\right )\right )}{18063360 c^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
Output:
(d^2*(-352800*b*(8*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 7056 00*a*Sqrt[d]*(8*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2* d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-2822400*b*c^2*f* g*x - 5160960*a*c*f*g*Sqrt[1 - c^2*x^2] + 12418560*a*c^3*f^2*x*Sqrt[1 - c^ 2*x^2] - 705600*a*c*g^2*x*Sqrt[1 - c^2*x^2] + 15482880*a*c^3*f*g*x^2*Sqrt[ 1 - c^2*x^2] - 9784320*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] + 5550720*a*c^3*g^2 *x^3*Sqrt[1 - c^2*x^2] - 15482880*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 301056 0*a*c^7*f^2*x^5*Sqrt[1 - c^2*x^2] - 6397440*a*c^5*g^2*x^5*Sqrt[1 - c^2*x^2 ] + 5160960*a*c^7*f*g*x^6*Sqrt[1 - c^2*x^2] + 2257920*a*c^7*g^2*x^7*Sqrt[1 - c^2*x^2] + 141120*b*(15*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] + 564480*b*c* f*g*Cos[3*ArcCos[c*x]] - 211680*b*c^2*f^2*Cos[4*ArcCos[c*x]] + 35280*b*g^2 *Cos[4*ArcCos[c*x]] - 112896*b*c*f*g*Cos[5*ArcCos[c*x]] + 15680*b*c^2*f^2* Cos[6*ArcCos[c*x]] - 15680*b*g^2*Cos[6*ArcCos[c*x]] + 11520*b*c*f*g*Cos[7* ArcCos[c*x]] + 2205*b*g^2*Cos[8*ArcCos[c*x]]) + 168*b*Sqrt[d - c^2*d*x^2]* ArcCos[c*x]*(-58112*c*f*g*Sqrt[1 - c^2*x^2] + 111872*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] - 27648*c*f*g*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 3840*c*f*g *(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 25200*c^2*f^2*Sin[2*ArcCos[c*x]] + 1680*g^2*Sin[2*ArcCos[c*x]] - 8960*c*f*g*Sin[3*ArcCos[c*x]] - 5040*c^2* f^2*Sin[4*ArcCos[c*x]] + 840*g^2*Sin[4*ArcCos[c*x]] - 5376*c*f*g*Sin[5*Arc Cos[c*x]] + 560*c^2*f^2*Sin[6*ArcCos[c*x]] - 560*g^2*Sin[6*ArcCos[c*x]]...
Time = 1.23 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.55, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^2 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f^2 (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{5/2}+g^2 x^2 (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{5/2}+2 f g x (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{5/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5 g^2 (a+b \arccos (c x))^2}{256 b c^3}+\frac {1}{6} f^2 x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{24} f^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {5}{16} f^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {2 f g \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}-\frac {5 g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{128 c^2}+\frac {1}{8} g^2 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{48} g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {5}{64} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {5 f^2 (a+b \arccos (c x))^2}{32 b c}+\frac {2}{49} b c^5 f g x^7+\frac {1}{64} b c^5 g^2 x^8-\frac {5}{96} b c^3 f^2 x^4-\frac {6}{35} b c^3 f g x^5-\frac {17}{288} b c^3 g^2 x^6-\frac {b f^2 \left (1-c^2 x^2\right )^3}{36 c}+\frac {25}{96} b c f^2 x^2+\frac {2}{7} b c f g x^3-\frac {2 b f g x}{7 c}+\frac {59}{768} b c g^2 x^4-\frac {5 b g^2 x^2}{256 c}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
Output:
(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*Sqr t[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/16 - (5*g^2*x*Sqrt[1 - c^2*x^2]*(a + b *ArcCos[c*x]))/(128*c^2) + (5*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x] ))/64 + (5*f^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/24 + (5*g^2*x^3* (1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/48 + (f^2*x*(1 - c^2*x^2)^(5/2)*( a + b*ArcCos[c*x]))/6 + (g^2*x^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/ 8 - (2*f*g*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/(7*c^2) - (5*f^2*(a + b*ArcCos[c*x])^2)/(32*b*c) - (5*g^2*(a + b*ArcCos[c*x])^2)/(256*b*c^3)))/S qrt[1 - c^2*x^2]
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 = 1.25 (sec) , antiderivative size = 2204, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2204\) |
| parts | \(\text {Expression too large to display}\) | \(2204\) |
Input:
int((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBO SE)
Output:
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*(-1/160*(-d*(c^2*x^2-1) )^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(3*I+5*arccos(c*x))*sin(4 *arccos(c*x))*d^2/c^2/(c^2*x^2-1)+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*c^9* x^9-320*c^7*x^7+128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+272*c^5*x^5-256*I*(-c^2*x ^2+1)^(1/2)*x^6*c^6-88*c^3*x^3+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c*x-32*I *(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*g^2*(I+8*arccos(c*x))*d^ 2/c^3/(c^2*x^2-1)+1/3136*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64 *I*(-c^2*x^2+1)^(1/2)*x^7*c^7+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)*c*x+1)* f*g*(I+7*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(3 2*c^7*x^7-64*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+38*c^3*x^3-48*I*(-c^2 *x^2+1)^(1/2)*x^4*c^4-6*c*x+18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1) ^(1/2))*(I*c^2*f^2+6*arccos(c*x)*c^2*f^2-I*g^2-6*arccos(c*x)*g^2)*d^2/c^3/ (c^2*x^2-1)-1/1024*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2* x^2+1)^(1/2)*x^4*c^4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1...
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="f ricas")
Output:
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)*arccos(c*x))*sqrt(-c^2*d *x^2 + d), x)
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \] Input:
integrate((g*x+f)**2*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x)),x)
Output:
Timed out
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="m axima")
Output:
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(sqrt(c* x + 1)*sqrt(-c*x + 1), c*x), x)
Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="g iac")
Output:
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. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2),x)
Output:
int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2), x)
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (840 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+105 \mathit {asin} \left (c x \right ) a \,g^{2}+448 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} f^{2} x^{5}+768 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} f g \,x^{6}+336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} g^{2} x^{7}-1456 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f^{2} x^{3}-2304 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f g \,x^{4}-952 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} g^{2} x^{5}+1848 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +2304 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+826 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}-768 \sqrt {-c^{2} x^{2}+1}\, a c f g -105 \sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{6}d x \right ) b \,c^{7} g^{2}+5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{5}d x \right ) b \,c^{7} f g +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{7} f^{2}-5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{5} g^{2}-10752 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{5} f g -5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{5} f^{2}+2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+5376 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{3} f g +2688 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{3} f^{2}+768 a c f g \right )}{2688 c^{3}} \] Input:
int((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x)),x)
Output:
(sqrt(d)*d**2*(840*asin(c*x)*a*c**2*f**2 + 105*asin(c*x)*a*g**2 + 448*sqrt ( - c**2*x**2 + 1)*a*c**7*f**2*x**5 + 768*sqrt( - c**2*x**2 + 1)*a*c**7*f* g*x**6 + 336*sqrt( - c**2*x**2 + 1)*a*c**7*g**2*x**7 - 1456*sqrt( - c**2*x **2 + 1)*a*c**5*f**2*x**3 - 2304*sqrt( - c**2*x**2 + 1)*a*c**5*f*g*x**4 - 952*sqrt( - c**2*x**2 + 1)*a*c**5*g**2*x**5 + 1848*sqrt( - c**2*x**2 + 1)* a*c**3*f**2*x + 2304*sqrt( - c**2*x**2 + 1)*a*c**3*f*g*x**2 + 826*sqrt( - c**2*x**2 + 1)*a*c**3*g**2*x**3 - 768*sqrt( - c**2*x**2 + 1)*a*c*f*g - 105 *sqrt( - c**2*x**2 + 1)*a*c*g**2*x + 2688*int(sqrt( - c**2*x**2 + 1)*acos( c*x)*x**6,x)*b*c**7*g**2 + 5376*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**5, x)*b*c**7*f*g + 2688*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*b*c**7*f **2 - 5376*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*b*c**5*g**2 - 1075 2*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c**5*f*g - 5376*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**5*f**2 + 2688*int(sqrt( - c**2*x* *2 + 1)*acos(c*x)*x**2,x)*b*c**3*g**2 + 5376*int(sqrt( - c**2*x**2 + 1)*ac os(c*x)*x,x)*b*c**3*f*g + 2688*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x)*b*c **3*f**2 + 768*a*c*f*g))/(2688*c**3)