\(\int \frac {1}{x^3 (c+d \sqrt {a+b x^2})^{3/2}} \, dx\) [284]

Optimal result

Integrand size = 23, antiderivative size = 211 \[ \int \frac {1}{x^3 \left (c+d \sqrt {a+b x^2}\right )^{3/2}} \, dx=\frac {3 b c d^2}{\left (c^2-a d^2\right )^2 \sqrt {c+d \sqrt {a+b x^2}}}-\frac {c-d \sqrt {a+b x^2}}{2 \left (c^2-a d^2\right ) x^2 \sqrt {c+d \sqrt {a+b x^2}}}-\frac {3 b d \text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c-\sqrt {a} d}}\right )}{4 \sqrt {a} \left (c-\sqrt {a} d\right )^{5/2}}+\frac {3 b d \text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c+\sqrt {a} d}}\right )}{4 \sqrt {a} \left (c+\sqrt {a} d\right )^{5/2}} \] Output:

3*b*c*d^2/(-a*d^2+c^2)^2/(c+d*(b*x^2+a)^(1/2))^(1/2)-1/2*(c-d*(b*x^2+a)^(1 
/2))/(-a*d^2+c^2)/x^2/(c+d*(b*x^2+a)^(1/2))^(1/2)-3/4*b*d*arctanh((c+d*(b* 
x^2+a)^(1/2))^(1/2)/(c-a^(1/2)*d)^(1/2))/a^(1/2)/(c-a^(1/2)*d)^(5/2)+3/4*b 
*d*arctanh((c+d*(b*x^2+a)^(1/2))^(1/2)/(c+a^(1/2)*d)^(1/2))/a^(1/2)/(c+a^( 
1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \left (c+d \sqrt {a+b x^2}\right )^{3/2}} \, dx=\frac {1}{4} \left (\frac {2 \left (-c^3+a c d^2+6 b c d^2 x^2+d \left (c^2-a d^2\right ) \sqrt {a+b x^2}\right )}{\left (c^2-a d^2\right )^2 x^2 \sqrt {c+d \sqrt {a+b x^2}}}-\frac {3 b d \arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c-\sqrt {a} d}}\right )}{\sqrt {a} \left (-c-\sqrt {a} d\right )^{5/2}}+\frac {3 b d \arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c+\sqrt {a} d}}\right )}{\sqrt {a} \left (-c+\sqrt {a} d\right )^{5/2}}\right ) \] Input:

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

Output:

((2*(-c^3 + a*c*d^2 + 6*b*c*d^2*x^2 + d*(c^2 - a*d^2)*Sqrt[a + b*x^2]))/(( 
c^2 - a*d^2)^2*x^2*Sqrt[c + d*Sqrt[a + b*x^2]]) - (3*b*d*ArcTan[Sqrt[c + d 
*Sqrt[a + b*x^2]]/Sqrt[-c - Sqrt[a]*d]])/(Sqrt[a]*(-c - Sqrt[a]*d)^(5/2)) 
+ (3*b*d*ArcTan[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[-c + Sqrt[a]*d]])/(Sqrt[a 
]*(-c + Sqrt[a]*d)^(5/2)))/4
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

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

Input:

int(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

int(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Fricas [F(-1)]

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

integrate(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/x**3/(c+d*(b*x**2+a)**(1/2))**(3/2),x)
 

Output:

Integral(1/(x**3*(c + d*sqrt(a + b*x**2))**(3/2)), x)
 

Maxima [F]

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

integrate(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4863 vs. \(2 (170) = 340\).

Time = 1.57 (sec) , antiderivative size = 4863, normalized size of antiderivative = 23.05 \[ \int \frac {1}{x^3 \left (c+d \sqrt {a+b x^2}\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")
 

Output:

-1/4*(8*b*c*d^4*arctan(sqrt(sqrt(b*x^2 + a)*d + c)*sgn((sqrt(b*x^2 + a)*d 
+ c)*d - c*d)/sqrt(c*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - c))/((a^2*d^4 
- 2*a*c^2*d^2 + c^4)*sqrt(c*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - c)) - ( 
(a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d)^2*a^(5/2)*b*c*d^6 - (a^3*d 
^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d)^2*a^(3/2)*b*c^3*d^4*sgn((sqrt(b* 
x^2 + a)*d + c)*d - c*d) - (a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d) 
^2*a^(3/2)*b*c^3*d^4 + (a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d)^2*( 
5*a^(5/2)*c*d^6 - 4*a^(3/2)*c^3*d^4)*b*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d 
) - 2*(3*a^5*c^2*d^12 - 5*a^4*c^4*d^10 + 2*a^3*c^6*d^8)*b*abs(a^3*d^7 - 3* 
a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 
2*(7*a^4*c^4*d^10 - 11*a^3*c^6*d^8 + 4*a^2*c^8*d^6)*b*abs(a^3*d^7 - 3*a^2* 
c^2*d^5 + 3*a*c^4*d^3 - c^6*d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - 2*(5 
*a^3*c^6*d^8 - 7*a^2*c^8*d^6 + 2*a*c^10*d^4)*b*abs(a^3*d^7 - 3*a^2*c^2*d^5 
 + 3*a*c^4*d^3 - c^6*d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + 2*(a^2*c^8* 
d^6 - a*c^10*d^4)*b*abs(a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d)*sgn 
((sqrt(b*x^2 + a)*d + c)*d - c*d) - (3*a^6*d^14 - 4*a^5*c^2*d^12 + a^4*c^4 
*d^10)*b*abs(a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d) + (5*a^5*c^2*d 
^12 - 4*a^4*c^4*d^10 - a^3*c^6*d^8)*b*abs(a^3*d^7 - 3*a^2*c^2*d^5 + 3*a*c^ 
4*d^3 - c^6*d) - (a^4*c^4*d^10 + 4*a^3*c^6*d^8 - 5*a^2*c^8*d^6)*b*abs(a^3* 
d^7 - 3*a^2*c^2*d^5 + 3*a*c^4*d^3 - c^6*d) - (a^3*c^6*d^8 - 4*a^2*c^8*d...
 

Mupad [F(-1)]

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

int(1/(x^3*(c + d*(a + b*x^2)^(1/2))^(3/2)),x)
 

Output:

int(1/(x^3*(c + d*(a + b*x^2)^(1/2))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (c+d \sqrt {a+b x^2}\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int(1/x^3/(c+d*(b*x^2+a)^(1/2))^(3/2),x)
 

Output:

( - 81*sqrt(a)*sqrt(sqrt(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqrt(a)*d - 
 c)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sqrt(a)*d 
- c)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d - c)*sqr 
t(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a 
 + b*x**2)*d + c)*b*d**3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + 
b*x**2)*d + c)*c**2*d - 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c) 
*a*c*d**2 - sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d**2*x**2 
 + 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a**2*d**4 + 
 2*a*b*d**4*x**2 - 4*a*c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*a**4*b*c* 
d**9*x**2 - 81*sqrt(a)*sqrt(sqrt(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqr 
t(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sq 
rt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d 
- c)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqr 
t(sqrt(a + b*x**2)*d + c)*b*d**3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(s 
qrt(a + b*x**2)*d + c)*c**2*d - 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2 
)*d + c)*a*c*d**2 - sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d 
**2*x**2 + 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a** 
2*d**4 + 2*a*b*d**4*x**2 - 4*a*c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*a 
**3*b**2*c*d**9*x**4 - 90*sqrt(a)*sqrt(sqrt(a)*d - c)*atan((sqrt(a + b*x** 
2)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a + b...