Integrand size = 20, antiderivative size = 394 \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=-\frac {2 d \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 e x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {2 d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {e \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c^3}+\frac {2 d \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c}-\frac {e \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{3/2} c^3} \] Output:
-2*d*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^(1/2)-2*e*x^2*(-c^2*x^2+1)^( 1/2)/b/c/(a+b*arccos(c*x))^(1/2)-1/2*e*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelS( 2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3-2*d*2^(1/2)* Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2 ))/b^(3/2)/c+1/2*e*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*( a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+1/2*e*2^(1/2)*Pi^(1/2)*Fresnel C(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(3/2)/c^3+2 *d*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1 /2))*sin(a/b)/b^(3/2)/c-1/2*e*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*( a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/b^(3/2)/c^3
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\frac {e^{-\frac {3 i a}{b}} \left (8 c^2 d e^{\frac {3 i a}{b}} \sqrt {1-c^2 x^2}+8 c^2 e e^{\frac {3 i a}{b}} x^2 \sqrt {1-c^2 x^2}+i \left (4 c^2 d+e\right ) e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-i \left (4 c^2 d+e\right ) e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arccos (c x))}{b}\right )+i \sqrt {3} e \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )-i \sqrt {3} e e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )\right )}{4 b c^3 \sqrt {a+b \arccos (c x)}} \] Input:
Integrate[(d + e*x^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
(8*c^2*d*E^(((3*I)*a)/b)*Sqrt[1 - c^2*x^2] + 8*c^2*e*E^(((3*I)*a)/b)*x^2*S qrt[1 - c^2*x^2] + I*(4*c^2*d + e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCo s[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] - I*(4*c^2*d + e)*E^( ((4*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c *x]))/b] + I*Sqrt[3]*e*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-3* I)*(a + b*ArcCos[c*x]))/b] - I*Sqrt[3]*e*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*Ar cCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b])/(4*b*c^3*E^(((3* I)*a)/b)*Sqrt[a + b*ArcCos[c*x]])
Time = 1.00 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5173 |
| \(\displaystyle \int \left (\frac {d}{(a+b \arccos (c x))^{3/2}}+\frac {e x^2}{(a+b \arccos (c x))^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {\frac {\pi }{2}} e \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {3 \pi }{2}} e \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {\pi }{2}} e \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {\sqrt {\frac {3 \pi }{2}} e \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {2 \sqrt {2 \pi } d \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {2 \pi } d \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {2 d \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}+\frac {2 e x^2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arccos (c x)}}\) |
Input:
Int[(d + e*x^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
(2*d*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) + (2*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) - (e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[ (Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (2*d*Sqrt[2 *Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^( 3/2)*c) - (e*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*A rcCos[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (e*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]* Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) - (2*d*Sqrt[2*Pi ]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2 )*c) - (e*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqr t[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3)
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])
Time = 0.20 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(-\frac {4 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d -4 \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d +\sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \cos \left (\frac {3 a}{b}\right ) e -\sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sin \left (\frac {3 a}{b}\right ) e +\sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) e -\sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) e +4 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) c^{2} d +e \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )+e \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right )}{2 c^{3} b \sqrt {a +b \arccos \left (c x \right )}}\) | \(459\) |
Input:
int((e*x^2+d)/(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2/c^3/b*(4*2^(1/2)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*Fresne lC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*cos(a/b)*c^2*d -4*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)*sin(a/b)*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)*cos(3*a/b)*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)*sin(3*a/b)*e+2^(1/2)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*a rccos(c*x))^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x)) ^(1/2)/b)*cos(a/b)*e-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)*sin(a/b )*e+4*sin(-(a+b*arccos(c*x))/b+a/b)*c^2*d+e*sin(-3*(a+b*arccos(c*x))/b+3*a /b)+e*sin(-(a+b*arccos(c*x))/b+a/b))/(a+b*arccos(c*x))^(1/2)
Exception generated. \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] 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)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)/(a+b*acos(c*x))**(3/2),x)
Output:
Integral((d + e*x**2)/(a + b*acos(c*x))**(3/2), x)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] 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)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(a+b*arccos(c*x))^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/(b*arccos(c*x) + a)^(3/2), x)
Timed out. \[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)/(a + b*acos(c*x))^(3/2),x)
Output:
int((d + e*x^2)/(a + b*acos(c*x))^(3/2), x)
\[ \int \frac {d+e x^2}{(a+b \arccos (c x))^{3/2}} \, dx=\text {too large to display} \] Input:
int((e*x^2+d)/(a+b*acos(c*x))^(3/2),x)
Output:
( - 4*sqrt(acos(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*acos(c*x)*b*c**2*d + 8* sqrt(acos(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*acos(c*x)*b*e + 4*acos(c*x)*i nt(sqrt(acos(c*x)*b + a)/(acos(c*x)**2*b**2*c**2*x**2 - acos(c*x)**2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c**2*x**2 - a**2),x)* a*b**2*c**3*d - 8*acos(c*x)*int(sqrt(acos(c*x)*b + a)/(acos(c*x)**2*b**2*c **2*x**2 - acos(c*x)**2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c*e - 4*acos(c*x)*int((sqrt(acos(c*x)* b + a)*x**4)/(acos(c*x)**2*b**2*c**2*x**2 - acos(c*x)**2*b**2 + 2*acos(c*x )*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**7* d + 8*acos(c*x)*int((sqrt(acos(c*x)*b + a)*x**4)/(acos(c*x)**2*b**2*c**2*x **2 - acos(c*x)**2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a* *2*c**2*x**2 - a**2),x)*a*b**2*c**5*e - 24*acos(c*x)*int((sqrt(acos(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3)/(acos(c*x)**2*b**2*c**2*x**2 - acos(c*x)**2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c **2*x**2 - a**2),x)*a*b**2*c**6*d + 4*acos(c*x)*int((sqrt(acos(c*x)*b + a) *sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x)/(acos(c*x)**2*b**2*c**2*x**2 - aco s(c*x)**2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c**2*x **2 - a**2),x)*b**3*c**4*d - 8*acos(c*x)*int((sqrt(acos(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x)/(acos(c*x)**2*b**2*c**2*x**2 - acos(c*x)* *2*b**2 + 2*acos(c*x)*a*b*c**2*x**2 - 2*acos(c*x)*a*b + a**2*c**2*x**2 ...