\(\int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx\) [143]

Optimal result

Integrand size = 30, antiderivative size = 79 \[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {2 (f x)^{5/2} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{5 f}-\frac {4 b c (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2} \] Output:

2/5*(f*x)^(5/2)*(a+b*arccos(c*x))*hypergeom([1/2, 5/4],[9/4],c^2*x^2)/f-4/ 
35*b*c*(f*x)^(7/2)*hypergeom([1, 7/4, 7/4],[9/4, 11/4],c^2*x^2)/f^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.96 \[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {f \sqrt {f x} \left (-8 \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right ) \left (3 a-3 a c^2 x^2+2 b c x \sqrt {1-c^2 x^2}+3 b \arccos (c x)-3 b c^2 x^2 \arccos (c x)+3 i a \sqrt {-\frac {1}{c}} c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right ),-1\right )+3 b \left (-1+c^2 x^2\right ) \arccos (c x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {5}{4},c^2 x^2\right )\right )+3 b c \pi x \sqrt {2-2 c^2 x^2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )\right )}{36 c^2 \sqrt {1-c^2 x^2} \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {Gamma}\left (\frac {7}{4}\right )} \] Input:

Integrate[((f*x)^(3/2)*(a + b*ArcCos[c*x]))/Sqrt[1 - c^2*x^2],x]
 

Output:

(f*Sqrt[f*x]*(-8*Gamma[5/4]*Gamma[7/4]*(3*a - 3*a*c^2*x^2 + 2*b*c*x*Sqrt[1 
 - c^2*x^2] + 3*b*ArcCos[c*x] - 3*b*c^2*x^2*ArcCos[c*x] + (3*I)*a*Sqrt[-c^ 
(-1)]*c*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[-c^(-1)]/Sq 
rt[x]], -1] + 3*b*(-1 + c^2*x^2)*ArcCos[c*x]*Hypergeometric2F1[3/4, 1, 5/4 
, c^2*x^2]) + 3*b*c*Pi*x*Sqrt[2 - 2*c^2*x^2]*HypergeometricPFQ[{3/4, 3/4, 
1}, {5/4, 7/4}, c^2*x^2]))/(36*c^2*Sqrt[1 - c^2*x^2]*Gamma[5/4]*Gamma[7/4] 
)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {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.

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

\(\Big \downarrow \) 5221

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

Input:

Int[((f*x)^(3/2)*(a + b*ArcCos[c*x]))/Sqrt[1 - c^2*x^2],x]
 

Output:

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

Defintions of rubi rules used

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]

\[\int \frac {\left (f x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]

Input:

int((f*x)^(3/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2),x)
 

Output:

int((f*x)^(3/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^(3/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral(-sqrt(-c^2*x^2 + 1)*(b*f*x*arccos(c*x) + a*f*x)*sqrt(f*x)/(c^2*x^ 
2 - 1), x)
 

Sympy [F]

\[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (f x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate((f*x)**(3/2)*(a+b*acos(c*x))/(-c**2*x**2+1)**(1/2),x)
 

Output:

Integral((f*x)**(3/2)*(a + b*acos(c*x))/sqrt(-(c*x - 1)*(c*x + 1)), x)
 

Maxima [F]

\[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^(3/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="m 
axima")
 

Output:

integrate((f*x)^(3/2)*(b*arccos(c*x) + a)/sqrt(-c^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^(3/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((f*x)^(3/2)*(b*arccos(c*x) + a)/sqrt(-c^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (f\,x\right )}^{3/2}}{\sqrt {1-c^2\,x^2}} \,d x \] Input:

int(((a + b*acos(c*x))*(f*x)^(3/2))/(1 - c^2*x^2)^(1/2),x)
 

Output:

int(((a + b*acos(c*x))*(f*x)^(3/2))/(1 - c^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(f x)^{3/2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {\sqrt {f}\, f \left (-6 \sqrt {x}\, \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b -6 \sqrt {x}\, \sqrt {-c^{2} x^{2}+1}\, a -4 \sqrt {x}\, b c x -3 \left (\int \frac {\sqrt {x}\, \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b -3 \left (\int \frac {\sqrt {x}\, \sqrt {-c^{2} x^{2}+1}}{c^{2} x^{3}-x}d x \right ) a \right )}{9 c^{2}} \] Input:

int((f*x)^(3/2)*(a+b*acos(c*x))/(-c^2*x^2+1)^(1/2),x)
 

Output:

(sqrt(f)*f*( - 6*sqrt(x)*sqrt( - c**2*x**2 + 1)*acos(c*x)*b - 6*sqrt(x)*sq 
rt( - c**2*x**2 + 1)*a - 4*sqrt(x)*b*c*x - 3*int((sqrt(x)*sqrt( - c**2*x** 
2 + 1)*acos(c*x))/(c**2*x**3 - x),x)*b - 3*int((sqrt(x)*sqrt( - c**2*x**2 
+ 1))/(c**2*x**3 - x),x)*a))/(9*c**2)