\(\int \frac {a+b \arccos (c x)}{(d+e x^2)^{5/2}} \, dx\) [104]

Optimal result

Integrand size = 20, antiderivative size = 146 \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \] Output:

-1/3*b*c*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(1/2)+1/3*x*(a+b*arccos( 
c*x))/d/(e*x^2+d)^(3/2)+2/3*x*(a+b*arccos(c*x))/d^2/(e*x^2+d)^(1/2)-2/3*b* 
arctan(e^(1/2)*(-c^2*x^2+1)^(1/2)/c/(e*x^2+d)^(1/2))/d^2/e^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\sqrt {d+e x^2} \left (\frac {a x}{3 d \left (d+e x^2\right )^2}+\frac {2 a x}{3 d^2 \left (d+e x^2\right )}\right )+\frac {b c x^2 \sqrt {\frac {d+e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b x \left (3 d+2 e x^2\right ) \arccos (c x)}{3 d^2 \left (d+e x^2\right )^{3/2}} \] Input:

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

Output:

-1/3*(b*c*Sqrt[1 - c^2*x^2])/(d*(c^2*d + e)*Sqrt[d + e*x^2]) + Sqrt[d + e* 
x^2]*((a*x)/(3*d*(d + e*x^2)^2) + (2*a*x)/(3*d^2*(d + e*x^2))) + (b*c*x^2* 
Sqrt[(d + e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/(3*d^ 
2*Sqrt[d + e*x^2]) + (b*x*(3*d + 2*e*x^2)*ArcCos[c*x])/(3*d^2*(d + e*x^2)^ 
(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5171, 27, 435, 87, 66, 218}

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.

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

\(\Big \downarrow \) 5171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 435

\(\displaystyle \frac {b c \int \frac {2 e x^2+3 d}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx^2}{6 d^2}+\frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {b c \left (2 \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {b c \left (4 \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}+\frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 x (a+b \arccos (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arccos (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c \left (-\frac {4 \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}-\frac {2 d \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 d^2}\)

Input:

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

Output:

(x*(a + b*ArcCos[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCos[c*x]) 
)/(3*d^2*Sqrt[d + e*x^2]) + (b*c*((-2*d*Sqrt[1 - c^2*x^2])/((c^2*d + e)*Sq 
rt[d + e*x^2]) - (4*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2]) 
])/(c*Sqrt[e])))/(6*d^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

rule 5171

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x])   u, x 
] + Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 
, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (122) = 244\).

Time = 0.19 (sec) , antiderivative size = 686, normalized size of antiderivative = 4.70 \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {-e} + e^{2}\right ) - 2 \, {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arccos \left (c x\right ) - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, -\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arccos \left (c x\right ) - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/6*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + 
 b*d*e^2)*x^2)*sqrt(-e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d 
*e - c^2*e^2)*x^2 - 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(-c^2*x^2 + 1)*sqrt( 
e*x^2 + d)*sqrt(-e) + e^2) - 2*(2*(a*c^2*d*e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2 
*e + a*d*e^2)*x + (2*(b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^2*d^2*e + b*d*e^2) 
*x)*arccos(c*x) - (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(-c^2*x^2 + 1))*sqrt(e*x 
^2 + d))/(c^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e 
^2 + d^3*e^3)*x^2), -1/3*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e 
 + 2*(b*c^2*d^2*e + b*d*e^2)*x^2)*sqrt(e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d 
- e)*sqrt(-c^2*x^2 + 1)*sqrt(e*x^2 + d)*sqrt(e)/(c^3*e^2*x^4 - c*d*e + (c^ 
3*d*e - c*e^2)*x^2)) - (2*(a*c^2*d*e^2 + a*e^3)*x^3 + 3*(a*c^2*d^2*e + a*d 
*e^2)*x + (2*(b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^2*d^2*e + b*d*e^2)*x)*arcc 
os(c*x) - (b*c*d*e^2*x^2 + b*c*d^2*e)*sqrt(-c^2*x^2 + 1))*sqrt(e*x^2 + d)) 
/(c^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3 
*e^3)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( 
arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/((e^2*x^4 + 2*d*e*x^2 + d^2)*sq 
rt(e*x^2 + d)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {e \,x^{2}+d}\, a d e x +2 \sqrt {e \,x^{2}+d}\, a \,e^{2} x^{3}-2 \sqrt {e}\, a \,d^{2}-4 \sqrt {e}\, a d e \,x^{2}-2 \sqrt {e}\, a \,e^{2} x^{4}+3 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{4} e +6 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{3} e^{2} x^{2}+3 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{2} e^{3} x^{4}}{3 d^{2} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(3*sqrt(d + e*x**2)*a*d*e*x + 2*sqrt(d + e*x**2)*a*e**2*x**3 - 2*sqrt(e)*a 
*d**2 - 4*sqrt(e)*a*d*e*x**2 - 2*sqrt(e)*a*e**2*x**4 + 3*int(acos(c*x)/(sq 
rt(d + e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2* 
x**4),x)*b*d**4*e + 6*int(acos(c*x)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e* 
x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)*b*d**3*e**2*x**2 + 3*int(a 
cos(c*x)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e 
*x**2)*e**2*x**4),x)*b*d**2*e**3*x**4)/(3*d**2*e*(d**2 + 2*d*e*x**2 + e**2 
*x**4))