Integrand size = 27, antiderivative size = 214 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^4 d^2}-\frac {3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}} \] Output:
-1/4*b*x^2*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+x^3*(a+b*arccos(c *x))/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x ))/c^4/d^2-3/4*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/b/c^5/d/(-c^2*d*x^2+ d)^(1/2)+1/2*b*(-c^2*x^2+1)^(1/2)*ln(-c^2*x^2+1)/c^5/d/(-c^2*d*x^2+d)^(1/2 )
Time = 0.46 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a x}{2 c^4 d^2}-\frac {a x}{c^4 d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{2 c^5 d^{3/2}}-\frac {b \left (-8 c x \arccos (c x)-\sqrt {1-c^2 x^2} \left (6 \arccos (c x)^2+\cos (2 \arccos (c x))-8 \log \left (\sqrt {1-c^2 x^2}\right )+2 \arccos (c x) \sin (2 \arccos (c x))\right )\right )}{8 c^5 d \sqrt {d \left (1-c^2 x^2\right )}} \] Input:
Integrate[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(3/2),x]
Output:
Sqrt[-(d*(-1 + c^2*x^2))]*((a*x)/(2*c^4*d^2) - (a*x)/(c^4*d^2*(-1 + c^2*x^ 2))) + (3*a*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2) )])/(2*c^5*d^(3/2)) - (b*(-8*c*x*ArcCos[c*x] - Sqrt[1 - c^2*x^2]*(6*ArcCos [c*x]^2 + Cos[2*ArcCos[c*x]] - 8*Log[Sqrt[1 - c^2*x^2]] + 2*ArcCos[c*x]*Si n[2*ArcCos[c*x]])))/(8*c^5*d*Sqrt[d*(1 - c^2*x^2)])
Time = 0.78 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5207, 243, 49, 2009, 5211, 15, 5153}
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.
| \(\displaystyle \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5207 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^3}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \frac {x^2}{1-c^2 x^2}dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {b \sqrt {1-c^2 x^2} \int xdx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle \frac {x^3 (a+b \arccos (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(3/2),x]
Output:
(x^3*(a + b*ArcCos[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (3*(-1/4*(b*x^2*Sq rt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*A rcCos[c*x]))/(2*c^2*d) - (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c^ 3*Sqrt[d - c^2*d*x^2])))/(c^2*d) + (b*Sqrt[1 - c^2*x^2]*(-(x^2/c^2) - Log[ 1 - c^2*x^2]/c^4))/(2*c*d*Sqrt[d - c^2*d*x^2])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp [b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.02
| method | result | size |
| default | \(-\frac {a \,x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {9 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{8 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (3 \arccos \left (c x \right )\right )}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\) | \(432\) |
| parts | \(-\frac {a \,x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {9 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{8 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (3 \arccos \left (c x \right )\right )}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\) | \(432\) |
Input:
int(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*a*x^3/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*a/c^4*x/d/(-c^2*d*x^2+d)^(1/2)-3 /2*a/c^4/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-3/4* b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^5/(c^2*x^2-1)*arccos(c*x )^2-I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^5/(c^2*x^2-1)*arcc os(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^5/(c^2*x^2-1)*ln ((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)-9/8*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2 *x^2-1)*arccos(c*x)*x+1/16*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*(- c^2*x^2+1)^(1/2)+1/8*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*arccos(c *x)*cos(3*arccos(c*x))-1/16*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*s in(3*arccos(c*x))
\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas" )
Output:
integral((b*x^4*arccos(c*x) + a*x^4)*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2 *c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**4*(a+b*acos(c*x))/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral(x**4*(a + b*acos(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima" )
Output:
-1/2*a*(x^3/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 3*x/(sqrt(-c^2*d*x^2 + d)*c^4*d ) + 3*arcsin(c*x)/(c^5*d^(3/2))) - b*integrate(x^4*arctan2(sqrt(c*x + 1)*s qrt(-c*x + 1), c*x)/((c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqr t(d)
Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (190) = 380\).
Time = 0.87 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.53 \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")
Output:
1/8*c*(8*b*arccos(c*x)/(sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^ (3/2)*c^3*d^(3/2)/x^3) + 6*sqrt(-c^2*x^2 + 1)*b*arccos(c*x)^2/((sqrt(-c^2* x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c*x) + 8*a/ (sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3) + 12*sqrt(-c^2*x^2 + 1)*a*arccos(c*x)/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c*x) - 8*sqrt(-c^2*x^2 + 1)*b*log(2 )/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^ 3)*c*x) - 4*sqrt(-c^2*x^2 + 1)*b*log(abs(-c^2*x^2 + 1))/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c*x) + sqrt(-c^2* x^2 + 1)*b/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d ^(3/2)/x^3)*c*x) - 12*(c^2*x^2 - 1)*b*arccos(c*x)/((sqrt(-c^2*x^2 + 1)*c^5 *d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c^2*x^2) + 6*(-c^2*x^2 + 1)^(3/2)*b*arccos(c*x)^2/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c^3*x^3) - 12*(c^2*x^2 - 1)*a/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c^2*x^2) + 12*( -c^2*x^2 + 1)^(3/2)*a*arccos(c*x)/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c ^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c^3*x^3) - 8*(-c^2*x^2 + 1)^(3/2)*b*log (2)/((sqrt(-c^2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/ x^3)*c^3*x^3) - 4*(-c^2*x^2 + 1)^(3/2)*b*log(abs(-c^2*x^2 + 1))/((sqrt(-c^ 2*x^2 + 1)*c^5*d^(3/2)/x + (-c^2*x^2 + 1)^(3/2)*c^3*d^(3/2)/x^3)*c^3*x^...
Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^4*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(3/2),x)
Output:
int((x^4*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(3/2), x)
\[ \int \frac {x^4 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b -2 \mathit {acos} \left (c x \right ) b \,c^{3} x^{3}+2 \mathit {acos} \left (c x \right ) b c x -6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a -4 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b c +\sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, b -2 a \,c^{3} x^{3}+6 a c x}{4 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d} \] Input:
int(x^4*(a+b*acos(c*x))/(-c^2*d*x^2+d)^(3/2),x)
Output:
(3*sqrt( - c**2*x**2 + 1)*acos(c*x)**2*b - 2*acos(c*x)*b*c**3*x**3 + 2*aco s(c*x)*b*c*x - 6*sqrt( - c**2*x**2 + 1)*asin(c*x)*a - 4*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c + sqrt( - c**2*x**2 + 1)*b*c**2*x**2 - sqrt( - c**2*x**2 + 1)* b - 2*a*c**3*x**3 + 6*a*c*x)/(4*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**5*d)