\(\int \sqrt {d x} (a+b \arccos (c x))^2 \, dx\) [211]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int \sqrt {d x} (a+b \arccos (c x))^2 \, dx=\frac {2 (d x)^{3/2} (a+b \arccos (c x))^2}{3 d}+\frac {8 b c (d x)^{5/2} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{15 d^2}+\frac {16 b^2 c^2 (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 d^3} \] Output:

2/3*(d*x)^(3/2)*(a+b*arccos(c*x))^2/d+8/15*b*c*(d*x)^(5/2)*(a+b*arccos(c*x 
))*hypergeom([1/2, 5/4],[9/4],c^2*x^2)/d^2+16/105*b^2*c^2*(d*x)^(7/2)*hype 
rgeom([1, 7/4, 7/4],[9/4, 11/4],c^2*x^2)/d^3
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.85 \[ \int \sqrt {d x} (a+b \arccos (c x))^2 \, dx=\frac {1}{27} \sqrt {d x} \left (\frac {2 \left (9 a^2 c x-8 b^2 c x-12 a b \sqrt {1-c^2 x^2}+18 a b c x \arccos (c x)-12 b^2 \sqrt {1-c^2 x^2} \arccos (c x)+9 b^2 c x \arccos (c x)^2+12 a b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )+12 b^2 \sqrt {1-c^2 x^2} \arccos (c x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {5}{4},c^2 x^2\right )\right )}{c}+\frac {3 \sqrt {2} b^2 \pi x \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right )}\right ) \] Input:

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

Output:

(Sqrt[d*x]*((2*(9*a^2*c*x - 8*b^2*c*x - 12*a*b*Sqrt[1 - c^2*x^2] + 18*a*b* 
c*x*ArcCos[c*x] - 12*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 9*b^2*c*x*ArcCos[ 
c*x]^2 + 12*a*b*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + 12*b^2*Sqrt[1 
- c^2*x^2]*ArcCos[c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2]))/c + (3*Sq 
rt[2]*b^2*Pi*x*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gam 
ma[5/4]*Gamma[7/4])))/27
 

Rubi [A] (verified)

Time = 0.41 (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.

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

\(\Big \downarrow \) 5139

\(\displaystyle \frac {4 b c \int \frac {(d x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 d}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))^2}{3 d}\)

\(\Big \downarrow \) 5221

\(\displaystyle \frac {4 b c \left (\frac {4 b c (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 d^2}+\frac {2 (d x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right ) (a+b \arccos (c x))}{5 d}\right )}{3 d}+\frac {2 (d x)^{3/2} (a+b \arccos (c x))^2}{3 d}\)

Input:

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

Output:

(2*(d*x)^(3/2)*(a + b*ArcCos[c*x])^2)/(3*d) + (4*b*c*((2*(d*x)^(5/2)*(a + 
b*ArcCos[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*d) + (4*b*c*( 
d*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*d^2 
)))/(3*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]

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

Input:

int((d*x)^(1/2)*(a+b*arccos(c*x))^2,x)
 

Output:

int((d*x)^(1/2)*(a+b*arccos(c*x))^2,x)
 

Fricas [F]

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

integrate((d*x)^(1/2)*(a+b*arccos(c*x))^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d*x)**(1/2)*(a+b*acos(c*x))**2,x)
 

Output:

Integral(sqrt(d*x)*(a + b*acos(c*x))**2, x)
 

Maxima [F]

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

integrate((d*x)^(1/2)*(a+b*arccos(c*x))^2,x, algorithm="maxima")
 

Output:

2/3*b^2*sqrt(d)*x^(3/2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 1/6 
*a^2*c^2*sqrt(d)*(4*x^(3/2)/c^2 + 6*arctan(sqrt(c)*sqrt(x))/c^(7/2) + 3*lo 
g((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(7/2)) + 6*a*b*c^2*sqrt(d 
)*integrate(1/3*x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^ 
2 - 1), x) - 4*b^2*c*sqrt(d)*integrate(1/3*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^ 
(3/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x) - 1/2*a 
^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/c^(3/2) + log((c*sqrt(x) - sqrt(c))/ 
(c*sqrt(x) + sqrt(c)))/c^(3/2)) - 6*a*b*sqrt(d)*integrate(1/3*sqrt(x)*arct 
an(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x)
 

Giac [F(-2)]

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

integrate((d*x)^(1/2)*(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. \[ \int \sqrt {d x} (a+b \arccos (c x))^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\sqrt {d\,x} \,d x \] Input:

int((a + b*acos(c*x))^2*(d*x)^(1/2),x)
 

Output:

int((a + b*acos(c*x))^2*(d*x)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {d x} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {x}\, a^{2} x +6 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right )d x \right ) a b +3 \left (\int \sqrt {x}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2}\right )}{3} \] Input:

int((d*x)^(1/2)*(a+b*acos(c*x))^2,x)
 

Output:

(sqrt(d)*(2*sqrt(x)*a**2*x + 6*int(sqrt(x)*acos(c*x),x)*a*b + 3*int(sqrt(x 
)*acos(c*x)**2,x)*b**2))/3