Integrand size = 29, antiderivative size = 684 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\frac {d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} d e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} d e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-2 n} d e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-2 n} d e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} 3^{-1-n} d e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} 3^{-1-n} d e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arccos (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}} \] Output:
1/16*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^(1+n)/b/c^3/(1+n)/(-c^2*x^2+ 1)^(1/2)-I*2^(-7-n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^n*GAMMA(1+n,- 2*I*(a+b*arccos(c*x))/b)/c^3/exp(2*I*a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b*arc cos(c*x))/b)^n)+I*2^(-7-n)*d*exp(2*I*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos (c*x))^n*GAMMA(1+n,2*I*(a+b*arccos(c*x))/b)/c^3/(-c^2*x^2+1)^(1/2)/((I*(a+ b*arccos(c*x))/b)^n)+I*2^(-7-2*n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)) ^n*GAMMA(1+n,-4*I*(a+b*arccos(c*x))/b)/c^3/exp(4*I*a/b)/(-c^2*x^2+1)^(1/2) /((-I*(a+b*arccos(c*x))/b)^n)-I*2^(-7-2*n)*d*exp(4*I*a/b)*(-c^2*d*x^2+d)^( 1/2)*(a+b*arccos(c*x))^n*GAMMA(1+n,4*I*(a+b*arccos(c*x))/b)/c^3/(-c^2*x^2+ 1)^(1/2)/((I*(a+b*arccos(c*x))/b)^n)+I*2^(-7-n)*3^(-1-n)*d*(-c^2*d*x^2+d)^ (1/2)*(a+b*arccos(c*x))^n*GAMMA(1+n,-6*I*(a+b*arccos(c*x))/b)/c^3/exp(6*I* a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b*arccos(c*x))/b)^n)-I*2^(-7-n)*3^(-1-n)*d *exp(6*I*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^n*GAMMA(1+n,6*I*(a+b* arccos(c*x))/b)/c^3/(-c^2*x^2+1)^(1/2)/((I*(a+b*arccos(c*x))/b)^n)
Time = 3.01 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.74 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\frac {d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^n \left (-\frac {24 a}{b+b n}-\frac {24 \arccos (c x)}{1+n}-3 i 2^{-n} e^{-\frac {2 i a}{b}} \left (\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arccos (c x))}{b}\right )+3 i 2^{-n} e^{\frac {2 i a}{b}} \left (-\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arccos (c x))}{b}\right )-3 i 4^{-n} e^{-\frac {4 i a}{b}} \left (\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arccos (c x))}{b}\right )+3 i 4^{-n} e^{\frac {4 i a}{b}} \left (-\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arccos (c x))}{b}\right )+i 6^{-n} e^{-\frac {6 i a}{b}} \left (\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arccos (c x))}{b}\right )-i 6^{-n} e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arccos (c x))}{b}\right )^n \left (\frac {(a+b \arccos (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arccos (c x))}{b}\right )\right )}{384 c^3 \sqrt {d-c^2 d x^2}} \] Input:
Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^n,x]
Output:
(d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^n*((-24*a)/(b + b*n) - (24*ArcC os[c*x])/(1 + n) - ((3*I)*((I*(a + b*ArcCos[c*x]))/b)^n*Gamma[1 + n, ((-2* I)*(a + b*ArcCos[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*((a + b*ArcCos[c*x])^2/b^ 2)^n) + ((3*I)*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcCos[c*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcCos[c*x]))/b])/(2^n*((a + b*ArcCos[c*x])^2/b^2)^n) - ( (3*I)*((I*(a + b*ArcCos[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcCos[c*x ]))/b])/(4^n*E^(((4*I)*a)/b)*((a + b*ArcCos[c*x])^2/b^2)^n) + ((3*I)*E^((( 4*I)*a)/b)*(((-I)*(a + b*ArcCos[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*Arc Cos[c*x]))/b])/(4^n*((a + b*ArcCos[c*x])^2/b^2)^n) + (I*((I*(a + b*ArcCos[ c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcCos[c*x]))/b])/(6^n*E^(((6*I)*a )/b)*((a + b*ArcCos[c*x])^2/b^2)^n) - (I*E^(((6*I)*a)/b)*(((-I)*(a + b*Arc Cos[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcCos[c*x]))/b])/(6^n*((a + b* ArcCos[c*x])^2/b^2)^n)))/(384*c^3*Sqrt[d - c^2*d*x^2])
Time = 0.91 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5225, 4906, 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.
\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int (a+b \arccos (c x))^n \cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )d(a+b \arccos (c x))}{b c^3 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{32} \cos \left (\frac {6 a}{b}-\frac {6 (a+b \arccos (c x))}{b}\right ) (a+b \arccos (c x))^n-\frac {1}{16} \cos \left (\frac {4 a}{b}-\frac {4 (a+b \arccos (c x))}{b}\right ) (a+b \arccos (c x))^n-\frac {1}{32} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right ) (a+b \arccos (c x))^n+\frac {1}{16} (a+b \arccos (c x))^n\right )d(a+b \arccos (c x))}{b c^3 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \arccos (c x))^{n+1}}{16 (n+1)}+i b 2^{-n-7} e^{-\frac {2 i a}{b}} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i (a+b \arccos (c x))}{b}\right )+i b 2^{-2 n-7} e^{-\frac {4 i a}{b}} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 i (a+b \arccos (c x))}{b}\right )-i b 2^{-n-7} 3^{-n-1} e^{-\frac {6 i a}{b}} (a+b \arccos (c x))^n \left (-\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 i (a+b \arccos (c x))}{b}\right )-i b 2^{-n-7} e^{\frac {2 i a}{b}} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i (a+b \arccos (c x))}{b}\right )-i b 2^{-2 n-7} e^{\frac {4 i a}{b}} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i (a+b \arccos (c x))}{b}\right )+i b 2^{-n-7} 3^{-n-1} e^{\frac {6 i a}{b}} (a+b \arccos (c x))^n \left (\frac {i (a+b \arccos (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 i (a+b \arccos (c x))}{b}\right )\right )}{b c^3 \sqrt {1-c^2 x^2}}\) |
Input:
Int[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^n,x]
Output:
-((d*Sqrt[d - c^2*d*x^2]*((a + b*ArcCos[c*x])^(1 + n)/(16*(1 + n)) + (I*2^ (-7 - n)*b*(a + b*ArcCos[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcCos[c*x])) /b])/(E^(((2*I)*a)/b)*(((-I)*(a + b*ArcCos[c*x]))/b)^n) - (I*2^(-7 - n)*b* E^(((2*I)*a)/b)*(a + b*ArcCos[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcCos[c* x]))/b])/((I*(a + b*ArcCos[c*x]))/b)^n + (I*2^(-7 - 2*n)*b*(a + b*ArcCos[c *x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcCos[c*x]))/b])/(E^(((4*I)*a)/b)*(((- I)*(a + b*ArcCos[c*x]))/b)^n) - (I*2^(-7 - 2*n)*b*E^(((4*I)*a)/b)*(a + b*A rcCos[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcCos[c*x]))/b])/((I*(a + b*ArcC os[c*x]))/b)^n - (I*2^(-7 - n)*3^(-1 - n)*b*(a + b*ArcCos[c*x])^n*Gamma[1 + n, ((-6*I)*(a + b*ArcCos[c*x]))/b])/(E^(((6*I)*a)/b)*(((-I)*(a + b*ArcCo s[c*x]))/b)^n) + (I*2^(-7 - n)*3^(-1 - n)*b*E^(((6*I)*a)/b)*(a + b*ArcCos[ c*x])^n*Gamma[1 + n, ((6*I)*(a + b*ArcCos[c*x]))/b])/((I*(a + b*ArcCos[c*x ]))/b)^n))/(b*c^3*Sqrt[1 - c^2*x^2]))
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{n}d x\]
Input:
int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^n,x)
Output:
int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^n,x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^n,x, algorithm="frica s")
Output:
integral(-(c^2*d*x^4 - d*x^2)*sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)^n, x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\text {Timed out} \] Input:
integrate(x**2*(-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))**n,x)
Output:
Timed out
\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^n,x, algorithm="maxim a")
Output:
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccos(c*x) + a)^n*x^2, x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^n,x, algorithm="giac" )
Output:
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccos(c*x) + a)^n*x^2, x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:
int(x^2*(a + b*acos(c*x))^n*(d - c^2*d*x^2)^(3/2),x)
Output:
int(x^2*(a + b*acos(c*x))^n*(d - c^2*d*x^2)^(3/2), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^n \, dx=\sqrt {d}\, d \left (-\left (\int \left (\mathit {acos} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{4}d x \right ) c^{2}+\int \left (\mathit {acos} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) \] Input:
int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))^n,x)
Output:
sqrt(d)*d*( - int((acos(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**4,x)*c**2 + int((acos(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**2,x))