\(\int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx\) [209]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}+\frac {8 b c (d x)^{9/2} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {13}{4},c^2 x^2\right )}{63 d^2}+\frac {16 b^2 c^2 (d x)^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};c^2 x^2\right )}{693 d^3} \] Output:

2/7*(d*x)^(7/2)*(a+b*arccos(c*x))^2/d+8/63*b*c*(d*x)^(9/2)*(a+b*arccos(c*x 
))*hypergeom([1/2, 9/4],[13/4],c^2*x^2)/d^2+16/693*b^2*c^2*(d*x)^(11/2)*hy 
pergeom([1, 11/4, 11/4],[13/4, 15/4],c^2*x^2)/d^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(109)=218\).

Time = 10.85 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {(d x)^{5/2} \left (882 a^2 x^3+\frac {84 a b \left (-2 \sqrt {1-c^2 x^2} \left (5+3 c^2 x^2\right )+21 c^3 x^3 \arccos (c x)+10 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )\right )}{c^3}+\frac {b^2 \left (-16 c x \left (35+9 c^2 x^2\right )-168 \sqrt {1-c^2 x^2} \left (5+3 c^2 x^2\right ) \arccos (c x)+882 c^3 x^3 \arccos (c x)^2+840 \sqrt {1-c^2 x^2} \arccos (c x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {5}{4},c^2 x^2\right )+\frac {105 \sqrt {2} c \pi x \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right )}\right )}{c^3}\right )}{3087 x^2} \] Input:

Integrate[(d*x)^(5/2)*(a + b*ArcCos[c*x])^2,x]
 

Output:

((d*x)^(5/2)*(882*a^2*x^3 + (84*a*b*(-2*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^2) 
+ 21*c^3*x^3*ArcCos[c*x] + 10*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2]))/ 
c^3 + (b^2*(-16*c*x*(35 + 9*c^2*x^2) - 168*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^ 
2)*ArcCos[c*x] + 882*c^3*x^3*ArcCos[c*x]^2 + 840*Sqrt[1 - c^2*x^2]*ArcCos[ 
c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2] + (105*Sqrt[2]*c*Pi*x*Hyperge 
ometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4])))/ 
c^3))/(3087*x^2)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5139, 5221}

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.

\(\displaystyle \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {4 b c \int \frac {(d x)^{7/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{7 d}+\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}\)

\(\Big \downarrow \) 5221

\(\displaystyle \frac {4 b c \left (\frac {4 b c (d x)^{11/2} \, _3F_2\left (1,\frac {11}{4},\frac {11}{4};\frac {13}{4},\frac {15}{4};c^2 x^2\right )}{99 d^2}+\frac {2 (d x)^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {13}{4},c^2 x^2\right ) (a+b \arccos (c x))}{9 d}\right )}{7 d}+\frac {2 (d x)^{7/2} (a+b \arccos (c x))^2}{7 d}\)

Input:

Int[(d*x)^(5/2)*(a + b*ArcCos[c*x])^2,x]
 

Output:

(2*(d*x)^(7/2)*(a + b*ArcCos[c*x])^2)/(7*d) + (4*b*c*((2*(d*x)^(9/2)*(a + 
b*ArcCos[c*x])*Hypergeometric2F1[1/2, 9/4, 13/4, c^2*x^2])/(9*d) + (4*b*c* 
(d*x)^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, c^2*x^2])/(9 
9*d^2)))/(7*d)
 

Defintions of rubi rules used

rule 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5221

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. 
)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*(a + b*ArcCos[c*x] 
)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*Hypergeometric2F1[1/2, (1 + m)/2, 
 (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S 
imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m 
/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \left (d x \right )^{\frac {5}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}d x\]

Input:

int((d*x)^(5/2)*(a+b*arccos(c*x))^2,x)
 

Output:

int((d*x)^(5/2)*(a+b*arccos(c*x))^2,x)
 

Fricas [F]

\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="fricas")
 

Output:

integral((b^2*d^2*x^2*arccos(c*x)^2 + 2*a*b*d^2*x^2*arccos(c*x) + a^2*d^2* 
x^2)*sqrt(d*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Timed out} \] Input:

integrate((d*x)**(5/2)*(a+b*acos(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="maxima")
 

Output:

2/7*b^2*d^(5/2)*x^(7/2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 1/4 
2*a^2*c^2*d^(5/2)*(4*(3*c^2*x^(7/2) + 7*x^(3/2))/c^4 + 42*arctan(sqrt(c)*s 
qrt(x))/c^(11/2) + 21*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^( 
11/2)) + 14*a*b*c^2*d^(5/2)*integrate(1/7*x^(9/2)*arctan(sqrt(c*x + 1)*sqr 
t(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x) - 4*b^2*c*d^(5/2)*integrate(1/7*sqrt( 
c*x + 1)*sqrt(-c*x + 1)*x^(7/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x)) 
/(c^2*x^2 - 1), x) - 1/6*a^2*d^(5/2)*(4*x^(3/2)/c^2 + 6*arctan(sqrt(c)*sqr 
t(x))/c^(7/2) + 3*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(7/2) 
) - 14*a*b*d^(5/2)*integrate(1/7*x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 
1)/(c*x))/(c^2*x^2 - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d*x)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="giac")
 

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
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^{5/2} \,d x \] Input:

int((a + b*acos(c*x))^2*(d*x)^(5/2),x)
 

Output:

int((a + b*acos(c*x))^2*(d*x)^(5/2), x)
 

Reduce [F]

\[ \int (d x)^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {x}\, a^{2} x^{3}+14 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b +7 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2}\right )}{7} \] Input:

int((d*x)^(5/2)*(a+b*acos(c*x))^2,x)
 

Output:

(sqrt(d)*d**2*(2*sqrt(x)*a**2*x**3 + 14*int(sqrt(x)*acos(c*x)*x**2,x)*a*b 
+ 7*int(sqrt(x)*acos(c*x)**2*x**2,x)*b**2))/7