Integrand size = 18, antiderivative size = 225 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \sqrt {1-c^2 x^2}}{35 c^7}+\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{105 c^7}-\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}+\frac {b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x)) \] Output:
-1/35*b*(35*c^6*d^3+35*c^4*d^2*e+21*c^2*d*e^2+5*e^3)*(-c^2*x^2+1)^(1/2)/c^ 7+1/105*b*e*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(-c^2*x^2+1)^(3/2)/c^7-3/175*b* e^2*(7*c^2*d+5*e)*(-c^2*x^2+1)^(5/2)/c^7+1/49*b*e^3*(-c^2*x^2+1)^(7/2)/c^7 +d^3*x*(a+b*arccos(c*x))+d^2*e*x^3*(a+b*arccos(c*x))+3/5*d*e^2*x^5*(a+b*ar ccos(c*x))+1/7*e^3*x^7*(a+b*arccos(c*x))
Time = 0.15 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.84 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=a \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right )-\frac {b \sqrt {1-c^2 x^2} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )}{3675 c^7}+b \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right ) \arccos (c x) \] Input:
Integrate[(d + e*x^2)^3*(a + b*ArcCos[c*x]),x]
Output:
a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[1 - c^2*x^ 2]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^ 2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3* x^6)))/(3675*c^7) + b*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7)* ArcCos[c*x]
Time = 0.60 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5171, 27, 2331, 2389, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5171 |
| \(\displaystyle b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{35 \sqrt {1-c^2 x^2}}dx+d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{\sqrt {1-c^2 x^2}}dx+d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{70} b c \int \frac {5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3}{\sqrt {1-c^2 x^2}}dx^2+d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {1}{70} b c \int \left (-\frac {5 \left (1-c^2 x^2\right )^{5/2} e^3}{c^6}+\frac {3 \left (7 d c^2+5 e\right ) \left (1-c^2 x^2\right )^{3/2} e^2}{c^6}-\frac {\left (35 d^2 c^4+42 d e c^2+15 e^2\right ) \sqrt {1-c^2 x^2} e}{c^6}+\frac {35 d^3 c^6+35 d^2 e c^4+21 d e^2 c^2+5 e^3}{c^6 \sqrt {1-c^2 x^2}}\right )dx^2+d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 x (a+b \arccos (c x))+d^2 e x^3 (a+b \arccos (c x))+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))+\frac {1}{70} b c \left (-\frac {6 e^2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+5 e\right )}{5 c^8}+\frac {10 e^3 \left (1-c^2 x^2\right )^{7/2}}{7 c^8}+\frac {2 e \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{3 c^8}-\frac {2 \sqrt {1-c^2 x^2} \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right )}{c^8}\right )\) |
Input:
Int[(d + e*x^2)^3*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*Sqrt[1 - c^2* x^2])/c^8 + (2*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^(3/2))/( 3*c^8) - (6*e^2*(7*c^2*d + 5*e)*(1 - c^2*x^2)^(5/2))/(5*c^8) + (10*e^3*(1 - c^2*x^2)^(7/2))/(7*c^8)))/70 + d^3*x*(a + b*ArcCos[c*x]) + d^2*e*x^3*(a + b*ArcCos[c*x]) + (3*d*e^2*x^5*(a + b*ArcCos[c*x]))/5 + (e^3*x^7*(a + b*A rcCos[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[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(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]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 , 0])
Time = 0.14 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.36
| method | result | size |
| parts | \(a \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b \left (\frac {c \arccos \left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \arccos \left (c x \right ) d \,e^{2} x^{5}}{5}+c \arccos \left (c x \right ) d^{2} e \,x^{3}+\arccos \left (c x \right ) d^{3} c x +\frac {5 e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )-35 d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+21 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+35 d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{35 c^{6}}\right )}{c}\) | \(305\) |
| derivativedivides | \(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\arccos \left (c x \right ) d^{3} c^{7} x +\arccos \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arccos \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )\right )}{c^{6}}}{c}\) | \(325\) |
| default | \(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\arccos \left (c x \right ) d^{3} c^{7} x +\arccos \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arccos \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )\right )}{c^{6}}}{c}\) | \(325\) |
| orering | \(\frac {x \left (325 e^{4} x^{8} c^{8}+1792 c^{8} d \,e^{3} x^{6}+4410 c^{8} d^{2} e^{2} x^{4}+30 c^{6} e^{4} x^{6}+9800 c^{8} d^{3} e \,x^{2}+294 c^{6} d \,e^{3} x^{4}+1225 c^{8} d^{4}+2450 c^{6} d^{2} e^{2} x^{2}+60 c^{4} e^{4} x^{4}-7350 c^{6} d^{3} e +1176 c^{4} d \,e^{3} x^{2}-4900 c^{4} d^{2} e^{2}+240 c^{2} e^{4} x^{2}-2352 c^{2} d \,e^{3}-480 e^{4}\right ) \left (a +b \arccos \left (c x \right )\right )}{1225 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (75 e^{3} x^{6} c^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 c^{6} d^{3}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) e x -\frac {\left (e \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3675 c^{8} \left (e \,x^{2}+d \right )^{3}}\) | \(360\) |
Input:
int((e*x^2+d)^3*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/7*e^3*x^7+3/5*d*e^2*x^5+d^2*e*x^3+d^3*x)+b/c*(1/7*c*arccos(c*x)*e^3*x ^7+3/5*c*arccos(c*x)*d*e^2*x^5+c*arccos(c*x)*d^2*e*x^3+arccos(c*x)*d^3*c*x +1/35/c^6*(5*e^3*(-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))-35*d^3*c ^6*(-c^2*x^2+1)^(1/2)+21*d*c^2*e^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)-8/15*(-c^2*x^2+1)^(1/2))+35*d^2*c^4*e*(-1/3*c^2* x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \arccos \left (c x\right ) - {\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \, {\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} + {\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^ 2*e*x^3 + 35*b*c^7*d^3*x)*arccos(c*x) - (75*b*c^6*e^3*x^6 + 3675*b*c^6*d^3 + 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 + 10*b*c^4*e^3) *x^4 + 240*b*e^3 + (1225*b*c^6*d^2*e + 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^ 2)*sqrt(-c^2*x^2 + 1))/c^7
Time = 0.63 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.75 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {acos}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {acos}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {acos}{\left (c x \right )}}{7} - \frac {b d^{3} \sqrt {- c^{2} x^{2} + 1}}{c} - \frac {b d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} - \frac {3 b d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {b e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {2 b d^{2} e \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} - \frac {4 b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} - \frac {6 b e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} - \frac {8 b e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{3} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**3*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*acos(c*x) + b*d**2*e*x**3*acos(c*x) + 3*b*d*e**2*x**5*acos(c*x)/5 + b*e**3*x**7*acos(c*x)/7 - b*d**3*sqrt(-c**2*x**2 + 1)/c - b*d**2*e*x**2 *sqrt(-c**2*x**2 + 1)/(3*c) - 3*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - b*e**3*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - 2*b*d**2*e*sqrt(-c**2*x**2 + 1 )/(3*c**3) - 4*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) - 6*b*e**3*x** 4*sqrt(-c**2*x**2 + 1)/(245*c**3) - 8*b*d*e**2*sqrt(-c**2*x**2 + 1)/(25*c* *5) - 8*b*e**3*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 16*b*e**3*sqrt(-c**2 *x**2 + 1)/(245*c**7), Ne(c, 0)), ((a + pi*b/2)*(d**3*x + d**2*e*x**3 + 3* d*e**2*x**5/5 + e**3*x**7/7), True))
Time = 0.12 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.32 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {1}{25} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arccos(c*x) - c *(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^2*e + 1/25*( 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*e^2 + 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^3 + a*d^3*x + (c* x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*b*d^3/c
Time = 0.14 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.41 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {1}{7} \, b e^{3} x^{7} \arccos \left (c x\right ) + \frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, b d e^{2} x^{5} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{3} x^{6}}{49 \, c} + \frac {3}{5} \, a d e^{2} x^{5} + b d^{2} e x^{3} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x^{4}}{25 \, c} + a d^{2} e x^{3} + b d^{3} x \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} e x^{2}}{3 \, c} - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x^{4}}{245 \, c^{3}} + a d^{3} x - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3}}{c} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x^{2}}{25 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e}{3 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x^{2}}{245 \, c^{5}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{25 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e^{3}}{245 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/7*b*e^3*x^7*arccos(c*x) + 1/7*a*e^3*x^7 + 3/5*b*d*e^2*x^5*arccos(c*x) - 1/49*sqrt(-c^2*x^2 + 1)*b*e^3*x^6/c + 3/5*a*d*e^2*x^5 + b*d^2*e*x^3*arccos (c*x) - 3/25*sqrt(-c^2*x^2 + 1)*b*d*e^2*x^4/c + a*d^2*e*x^3 + b*d^3*x*arcc os(c*x) - 1/3*sqrt(-c^2*x^2 + 1)*b*d^2*e*x^2/c - 6/245*sqrt(-c^2*x^2 + 1)* b*e^3*x^4/c^3 + a*d^3*x - sqrt(-c^2*x^2 + 1)*b*d^3/c - 4/25*sqrt(-c^2*x^2 + 1)*b*d*e^2*x^2/c^3 - 2/3*sqrt(-c^2*x^2 + 1)*b*d^2*e/c^3 - 8/245*sqrt(-c^ 2*x^2 + 1)*b*e^3*x^2/c^5 - 8/25*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^5 - 16/245*sq rt(-c^2*x^2 + 1)*b*e^3/c^7
Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int((a + b*acos(c*x))*(d + e*x^2)^3,x)
Output:
int((a + b*acos(c*x))*(d + e*x^2)^3, x)
Time = 0.24 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.50 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {3675 \mathit {acos} \left (c x \right ) b \,c^{7} d^{3} x +3675 \mathit {acos} \left (c x \right ) b \,c^{7} d^{2} e \,x^{3}+2205 \mathit {acos} \left (c x \right ) b \,c^{7} d \,e^{2} x^{5}+525 \mathit {acos} \left (c x \right ) b \,c^{7} e^{3} x^{7}-3675 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{3}-1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{2} e \,x^{2}-441 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d \,e^{2} x^{4}-75 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e^{3} x^{6}-2450 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2} e -588 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,e^{2} x^{2}-90 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{3} x^{4}-1176 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d \,e^{2}-120 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{3} x^{2}-240 \sqrt {-c^{2} x^{2}+1}\, b \,e^{3}+3675 a \,c^{7} d^{3} x +3675 a \,c^{7} d^{2} e \,x^{3}+2205 a \,c^{7} d \,e^{2} x^{5}+525 a \,c^{7} e^{3} x^{7}}{3675 c^{7}} \] Input:
int((e*x^2+d)^3*(a+b*acos(c*x)),x)
Output:
(3675*acos(c*x)*b*c**7*d**3*x + 3675*acos(c*x)*b*c**7*d**2*e*x**3 + 2205*a cos(c*x)*b*c**7*d*e**2*x**5 + 525*acos(c*x)*b*c**7*e**3*x**7 - 3675*sqrt( - c**2*x**2 + 1)*b*c**6*d**3 - 1225*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*e*x **2 - 441*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*x**4 - 75*sqrt( - c**2*x**2 + 1)*b*c**6*e**3*x**6 - 2450*sqrt( - c**2*x**2 + 1)*b*c**4*d**2*e - 588*s qrt( - c**2*x**2 + 1)*b*c**4*d*e**2*x**2 - 90*sqrt( - c**2*x**2 + 1)*b*c** 4*e**3*x**4 - 1176*sqrt( - c**2*x**2 + 1)*b*c**2*d*e**2 - 120*sqrt( - c**2 *x**2 + 1)*b*c**2*e**3*x**2 - 240*sqrt( - c**2*x**2 + 1)*b*e**3 + 3675*a*c **7*d**3*x + 3675*a*c**7*d**2*e*x**3 + 2205*a*c**7*d*e**2*x**5 + 525*a*c** 7*e**3*x**7)/(3675*c**7)