\(\int (d+e x^2) (a+b \arccos (c x))^{3/2} \, dx\) [693]

Optimal result

Integrand size = 20, antiderivative size = 482 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^{3/2} \, dx=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{6 c}+d x (a+b \arccos (c x))^{3/2}+\frac {1}{3} e x^3 (a+b \arccos (c x))^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \] Output:

3/2*b*d*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^(1/2)/c+1/3*b*e*(-c^2*x^2+1)^ 
(1/2)*(a+b*arccos(c*x))^(1/2)/c^3+1/6*b*e*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcc 
os(c*x))^(1/2)/c+d*x*(a+b*arccos(c*x))^(3/2)+1/3*e*x^3*(a+b*arccos(c*x))^( 
3/2)-3/4*b^(3/2)*d*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+ 
b*arccos(c*x))^(1/2)/b^(1/2))/c-3/16*b^(3/2)*e*2^(1/2)*Pi^(1/2)*cos(a/b)*F 
resnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/c^3+1/144*b^(3/2 
)*e*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x) 
)^(1/2)/b^(1/2))/c^3-3/4*b^(3/2)*d*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1 
/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)/c-3/16*b^(3/2)*e*2^(1/2)*Pi^ 
(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)/ 
c^3+1/144*b^(3/2)*e*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arccos 
(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.27 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.77 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^{3/2} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x^2)*(a + b*ArcCos[c*x])^(3/2),x]
 

Output:

((-1/2*I)*a*b*d*(-(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a 
+ b*ArcCos[c*x]))/b]) + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Ga 
mma[3/2, (I*(a + b*ArcCos[c*x]))/b]))/(c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x 
]]) - ((I/72)*a*b*e*(-9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b] 
*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + 
 b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(-(Sqr 
t[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] 
) + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + 
 b*ArcCos[c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]]) - (Sq 
rt[b]*d*(2*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*Ar 
cCos[c*x]) - Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt 
[b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[ 
a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c) - (Sqrt 
[b]*e*(18*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*Arc 
Cos[c*x]) - 9*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqr 
t[b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sq 
rt[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) - Sqrt[6*Pi] 
*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(b*Cos[(3*a)/b] + 
2*a*Sin[(3*a)/b]) - Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x] 
])/Sqrt[b]]*(2*a*Cos[(3*a)/b] - b*Sin[(3*a)/b]) + 6*Sqrt[b]*Sqrt[a + b*...
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5173, 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.

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

\(\Big \downarrow \) 5173

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{8 c^3}-\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}-\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{6 c}-\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^3}+d x (a+b \arccos (c x))^{3/2}+\frac {1}{3} e x^3 (a+b \arccos (c x))^{3/2}\)

Input:

Int[(d + e*x^2)*(a + b*ArcCos[c*x])^(3/2),x]
 

Output:

(-3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(2*c) - (b*e*Sqrt[1 - c 
^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(3*c^3) - (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt 
[a + b*ArcCos[c*x]])/(6*c) + d*x*(a + b*ArcCos[c*x])^(3/2) + (e*x^3*(a + b 
*ArcCos[c*x])^(3/2))/3 + (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2 
/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(2*c) + (3*b^(3/2)*e*Sqrt[Pi/2]*Co 
s[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(8*c^3) + ( 
b^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c 
*x]])/Sqrt[b]])/(24*c^3) - (3*b^(3/2)*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sq 
rt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]* 
FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) - 
 (b^(3/2)*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[ 
b]]*Sin[(3*a)/b])/(24*c^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5173

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(850\) vs. \(2(374)=748\).

Time = 0.22 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.77

method	result	size
default	\(\text {Expression too large to display}\)	\(851\)

Input:

int((e*x^2+d)*(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/144/c^3*(-108*2^(1/2)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*Fres 
nelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*sin(a/b)*b^2 
*c^2*d-108*2^(1/2)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelS( 
2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*cos(a/b)*b^2*c^2* 
d-2^(1/2)*(-3/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelC(3*2^(1/2) 
/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*sin(3*a/b)*b^2*e-2^(1/2) 
*(-3/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelS(3*2^(1/2)/Pi^(1/2) 
/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*cos(3*a/b)*b^2*e-27*2^(1/2)*(-1/b 
)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^ 
(1/2)*(a+b*arccos(c*x))^(1/2)/b)*sin(a/b)*b^2*e-27*2^(1/2)*(-1/b)^(1/2)*Pi 
^(1/2)*(a+b*arccos(c*x))^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b 
*arccos(c*x))^(1/2)/b)*cos(a/b)*b^2*e+144*arccos(c*x)^2*cos(-(a+b*arccos(c 
*x))/b+a/b)*b^2*c^2*d+216*arccos(c*x)*sin(-(a+b*arccos(c*x))/b+a/b)*b^2*c^ 
2*d+288*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*a*b*c^2*d+12*arccos(c*x) 
^2*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*b^2*e+36*arccos(c*x)^2*cos(-(a+b*arcc 
os(c*x))/b+a/b)*b^2*e+216*sin(-(a+b*arccos(c*x))/b+a/b)*a*b*c^2*d+144*cos( 
-(a+b*arccos(c*x))/b+a/b)*a^2*c^2*d+24*arccos(c*x)*cos(-3*(a+b*arccos(c*x) 
)/b+3*a/b)*a*b*e+6*arccos(c*x)*sin(-3*(a+b*arccos(c*x))/b+3*a/b)*b^2*e+54* 
arccos(c*x)*sin(-(a+b*arccos(c*x))/b+a/b)*b^2*e+72*arccos(c*x)*cos(-(a+b*a 
rccos(c*x))/b+a/b)*a*b*e+12*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*a^2*e+6*s...
 

Fricas [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x^2+d)*(a+b*arccos(c*x))^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate((e*x**2+d)*(a+b*acos(c*x))**(3/2),x)
 

Output:

Integral((a + b*acos(c*x))**(3/2)*(d + e*x**2), x)
 

Maxima [F]

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

integrate((e*x^2+d)*(a+b*arccos(c*x))^(3/2),x, algorithm="maxima")
 

Output:

integrate((e*x^2 + d)*(b*arccos(c*x) + a)^(3/2), x)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 3270, normalized size of antiderivative = 6.78 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)*(a+b*arccos(c*x))^(3/2),x, algorithm="giac")
 

Output:

-1/96*(96*I*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arccos( 
c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/ 
b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*sqrt(2)*sqrt(pi)*a 
*b^2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*s 
qrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b 
)) + b*sqrt(abs(b))) - 96*I*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(1/2*I*sqrt(2) 
*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a 
)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*s 
qrt(2)*sqrt(pi)*a*b^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt 
(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/ 
(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*I*sqrt(2)*sqrt(pi)*a^2*c^2*d*e 
rf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt( 
b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs( 
b))) + 48*sqrt(2)*sqrt(pi)*a*b*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) 
 + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e 
^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 36*I*sqrt(2)*sqrt(pi)*b^2*c^2 
*d*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*s 
qrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt( 
abs(b))) + 48*I*sqrt(2)*sqrt(pi)*a^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arccos 
(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(...
 

Mupad [F(-1)]

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

int((a + b*acos(c*x))^(3/2)*(d + e*x^2),x)
 

Output:

int((a + b*acos(c*x))^(3/2)*(d + e*x^2), x)
 

Reduce [F]

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

int((e*x^2+d)*(a+b*acos(c*x))^(3/2),x)
 

Output:

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