Integrand size = 19, antiderivative size = 152 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {b \left (7 c^2 d+5 e\right ) \sqrt {1-c^2 x^2}}{35 c^7}-\frac {b \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{3/2}}{105 c^7}+\frac {b \left (7 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x)) \] Output:
1/35*b*(7*c^2*d+5*e)*(-c^2*x^2+1)^(1/2)/c^7-1/105*b*(14*c^2*d+15*e)*(-c^2* x^2+1)^(3/2)/c^7+1/175*b*(7*c^2*d+15*e)*(-c^2*x^2+1)^(5/2)/c^7-1/49*b*e*(- c^2*x^2+1)^(7/2)/c^7+1/5*d*x^5*(a+b*arccos(c*x))+1/7*e*x^7*(a+b*arccos(c*x ))
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{5} a d x^5+\frac {1}{7} a e x^7+b d \sqrt {1-c^2 x^2} \left (-\frac {8}{75 c^5}-\frac {4 x^2}{75 c^3}-\frac {x^4}{25 c}\right )+b e \sqrt {1-c^2 x^2} \left (-\frac {16}{245 c^7}-\frac {8 x^2}{245 c^5}-\frac {6 x^4}{245 c^3}-\frac {x^6}{49 c}\right )+\frac {1}{5} b d x^5 \arccos (c x)+\frac {1}{7} b e x^7 \arccos (c x) \] Input:
Integrate[x^4*(d + e*x^2)*(a + b*ArcCos[c*x]),x]
Output:
(a*d*x^5)/5 + (a*e*x^7)/7 + b*d*Sqrt[1 - c^2*x^2]*(-8/(75*c^5) - (4*x^2)/( 75*c^3) - x^4/(25*c)) + b*e*Sqrt[1 - c^2*x^2]*(-16/(245*c^7) - (8*x^2)/(24 5*c^5) - (6*x^4)/(245*c^3) - x^6/(49*c)) + (b*d*x^5*ArcCos[c*x])/5 + (b*e* x^7*ArcCos[c*x])/7
Time = 0.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5231, 27, 354, 86, 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.
| \(\displaystyle \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle b c \int \frac {x^5 \left (5 e x^2+7 d\right )}{35 \sqrt {1-c^2 x^2}}dx+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} b c \int \frac {x^5 \left (5 e x^2+7 d\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{70} b c \int \frac {x^4 \left (5 e x^2+7 d\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{70} b c \int \left (-\frac {5 e \left (1-c^2 x^2\right )^{5/2}}{c^6}+\frac {\left (7 d c^2+15 e\right ) \left (1-c^2 x^2\right )^{3/2}}{c^6}+\frac {\left (-14 d c^2-15 e\right ) \sqrt {1-c^2 x^2}}{c^6}+\frac {7 d c^2+5 e}{c^6 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{7} e x^7 (a+b \arccos (c x))+\frac {1}{70} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+15 e\right )}{5 c^8}+\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (14 c^2 d+15 e\right )}{3 c^8}-\frac {2 \sqrt {1-c^2 x^2} \left (7 c^2 d+5 e\right )}{c^8}+\frac {10 e \left (1-c^2 x^2\right )^{7/2}}{7 c^8}\right )\) |
Input:
Int[x^4*(d + e*x^2)*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(7*c^2*d + 5*e)*Sqrt[1 - c^2*x^2])/c^8 + (2*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(3/2))/(3*c^8) - (2*(7*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(5*c ^8) + (10*e*(1 - c^2*x^2)^(7/2))/(7*c^8)))/70 + (d*x^5*(a + b*ArcCos[c*x]) )/5 + (e*x^7*(a + b*ArcCos[c*x]))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcCos[c*x]) u, x] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Time = 0.41 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.28
| method | result | size |
| parts | \(a \left (\frac {1}{7} e \,x^{7}+\frac {1}{5} d \,x^{5}\right )+\frac {b \left (\frac {c^{5} \arccos \left (c x \right ) e \,x^{7}}{7}+\frac {\arccos \left (c x \right ) c^{5} x^{5} d}{5}+\frac {5 e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )+7 d \,c^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{35 c^{2}}\right )}{c^{5}}\) | \(194\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\arccos \left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {d \,c^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{2}}}{c^{5}}\) | \(201\) |
| default | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\arccos \left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {d \,c^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{2}}}{c^{5}}\) | \(201\) |
| orering | \(\frac {\left (975 c^{8} e^{2} x^{10}+2442 c^{8} d e \,x^{8}+1323 c^{8} d^{2} x^{6}+90 x^{8} e^{2} c^{6}+354 x^{6} e \,c^{6} d +196 c^{6} d^{2} x^{4}+180 e^{2} x^{6} c^{4}+1296 c^{4} d e \,x^{4}+784 c^{4} d^{2} x^{2}+720 c^{2} e^{2} x^{4}-1872 c^{2} d e \,x^{2}-1568 c^{2} d^{2}-1440 e^{2} x^{2}-960 d e \right ) \left (a +b \arccos \left (c x \right )\right )}{3675 \left (e \,x^{2}+d \right ) x \,c^{8}}-\frac {\left (75 c^{6} e \,x^{6}+147 c^{6} d \,x^{4}+90 c^{4} e \,x^{4}+196 c^{4} d \,x^{2}+120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 x^{3} \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )+2 e \,x^{5} \left (a +b \arccos \left (c x \right )\right )-\frac {x^{4} \left (e \,x^{2}+d \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3675 c^{8} x^{4} \left (e \,x^{2}+d \right )}\) | \(308\) |
Input:
int(x^4*(e*x^2+d)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/7*e*x^7+1/5*d*x^5)+b/c^5*(1/7*c^5*arccos(c*x)*e*x^7+1/5*arccos(c*x)*c ^5*x^5*d+1/35/c^2*(5*e*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2 *x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^2*x^2+1)^(1/2))+7* d*c^2*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/1 5*(-c^2*x^2+1)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \arccos \left (c x\right ) - {\left (75 \, b c^{6} e x^{6} + 3 \, {\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \, {\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{7}} \] Input:
integrate(x^4*(e*x^2+d)*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/3675*(525*a*c^7*e*x^7 + 735*a*c^7*d*x^5 + 105*(5*b*c^7*e*x^7 + 7*b*c^7*d *x^5)*arccos(c*x) - (75*b*c^6*e*x^6 + 3*(49*b*c^6*d + 30*b*c^4*e)*x^4 + 39 2*b*c^2*d + 4*(49*b*c^4*d + 30*b*c^2*e)*x^2 + 240*b*e)*sqrt(-c^2*x^2 + 1)) /c^7
Time = 0.74 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.50 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {acos}{\left (c x \right )}}{7} - \frac {b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {b e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {4 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {6 b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b d \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} - \frac {8 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d x^{5}}{5} + \frac {e x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(e*x**2+d)*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d*x**5/5 + a*e*x**7/7 + b*d*x**5*acos(c*x)/5 + b*e*x**7*acos( c*x)/7 - b*d*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - b*e*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - 4*b*d*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) - 6*b*e*x**4*sqrt(- c**2*x**2 + 1)/(245*c**3) - 8*b*d*sqrt(-c**2*x**2 + 1)/(75*c**5) - 8*b*e*x **2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 16*b*e*sqrt(-c**2*x**2 + 1)/(245*c** 7), Ne(c, 0)), ((a + pi*b/2)*(d*x**5/5 + e*x**7/7), True))
Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d + \frac {1}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e \] Input:
integrate(x^4*(e*x^2+d)*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/7*a*e*x^7 + 1/5*a*d*x^5 + 1/75*(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b *d + 1/245*(35*x^7*arccos(c*x) - (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c ^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1) /c^8)*c)*b*e
Time = 0.13 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{7} \, b e x^{7} \arccos \left (c x\right ) + \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, b d x^{5} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e x^{6}}{49 \, c} + \frac {1}{5} \, a d x^{5} - \frac {\sqrt {-c^{2} x^{2} + 1} b d x^{4}}{25 \, c} - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b e x^{4}}{245 \, c^{3}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d x^{2}}{75 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e x^{2}}{245 \, c^{5}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d}{75 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e}{245 \, c^{7}} \] Input:
integrate(x^4*(e*x^2+d)*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/7*b*e*x^7*arccos(c*x) + 1/7*a*e*x^7 + 1/5*b*d*x^5*arccos(c*x) - 1/49*sqr t(-c^2*x^2 + 1)*b*e*x^6/c + 1/5*a*d*x^5 - 1/25*sqrt(-c^2*x^2 + 1)*b*d*x^4/ c - 6/245*sqrt(-c^2*x^2 + 1)*b*e*x^4/c^3 - 4/75*sqrt(-c^2*x^2 + 1)*b*d*x^2 /c^3 - 8/245*sqrt(-c^2*x^2 + 1)*b*e*x^2/c^5 - 8/75*sqrt(-c^2*x^2 + 1)*b*d/ c^5 - 16/245*sqrt(-c^2*x^2 + 1)*b*e/c^7
Timed out. \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \] Input:
int(x^4*(a + b*acos(c*x))*(d + e*x^2),x)
Output:
int(x^4*(a + b*acos(c*x))*(d + e*x^2), x)
Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.26 \[ \int x^4 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {735 \mathit {acos} \left (c x \right ) b \,c^{7} d \,x^{5}+525 \mathit {acos} \left (c x \right ) b \,c^{7} e \,x^{7}-147 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d \,x^{4}-75 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e \,x^{6}-196 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,x^{2}-90 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e \,x^{4}-392 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d -120 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e \,x^{2}-240 \sqrt {-c^{2} x^{2}+1}\, b e +735 a \,c^{7} d \,x^{5}+525 a \,c^{7} e \,x^{7}}{3675 c^{7}} \] Input:
int(x^4*(e*x^2+d)*(a+b*acos(c*x)),x)
Output:
(735*acos(c*x)*b*c**7*d*x**5 + 525*acos(c*x)*b*c**7*e*x**7 - 147*sqrt( - c **2*x**2 + 1)*b*c**6*d*x**4 - 75*sqrt( - c**2*x**2 + 1)*b*c**6*e*x**6 - 19 6*sqrt( - c**2*x**2 + 1)*b*c**4*d*x**2 - 90*sqrt( - c**2*x**2 + 1)*b*c**4* e*x**4 - 392*sqrt( - c**2*x**2 + 1)*b*c**2*d - 120*sqrt( - c**2*x**2 + 1)* b*c**2*e*x**2 - 240*sqrt( - c**2*x**2 + 1)*b*e + 735*a*c**7*d*x**5 + 525*a *c**7*e*x**7)/(3675*c**7)