3.30.43 \(\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx\) [2943]

3.30.43.1 Optimal result

Integrand size = 43, antiderivative size = 351 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx=\frac {\left (5 b-4 a^2 x^2\right ) \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{8 b}+\frac {a x \sqrt {-b+a^2 x^2} \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{2 b}-\frac {7 b^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt {3} \sqrt [3]{b}}\right )}{4\ 2^{2/3} \sqrt {3} a^{2/3}}-\frac {7 b^{2/3} \log \left (-1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}\right )}{12\ 2^{2/3} a^{2/3}}+\frac {7 b^{2/3} \log \left (1+\frac {\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}+\frac {2^{2/3} a^{2/3} \left (a x^2+x \sqrt {-b+a^2 x^2}\right )^{2/3}}{b^{2/3}}\right )}{24\ 2^{2/3} a^{2/3}} \]

output

1/8*(-4*a^2*x^2+5*b)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(2/3)/b+1/2*a*x*(a^2*x^2- 
b)^(1/2)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(2/3)/b-7/24*b^(2/3)*arctan(1/3*3^(1/ 
2)+2/3*2^(1/3)*a^(1/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/3)*3^(1/2)/b^(1/3))* 
2^(1/3)*3^(1/2)/a^(2/3)-7/24*b^(2/3)*ln(-1+2^(1/3)*a^(1/3)*(a*x^2+x*(a^2*x 
^2-b)^(1/2))^(1/3)/b^(1/3))*2^(1/3)/a^(2/3)+7/48*b^(2/3)*ln(1+2^(1/3)*a^(1 
/3)*(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/3)/b^(1/3)+2^(2/3)*a^(2/3)*(a*x^2+x*(a^ 
2*x^2-b)^(1/2))^(2/3)/b^(2/3))*2^(1/3)/a^(2/3)
 

3.30.43.2 Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx=\frac {\sqrt [3]{x} \left (-\frac {12 b x^{2/3}}{a x+\sqrt {-b+a^2 x^2}}+18 x^{2/3} \left (a x+\sqrt {-b+a^2 x^2}\right )-\frac {14 \sqrt [3]{2} \sqrt {3} b^{2/3} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {14 \sqrt [3]{2} b^{2/3} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \log \left (-\sqrt [3]{b}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )}{a^{2/3}}+\frac {7 \sqrt [3]{2} b^{2/3} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \log \left (b^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+2^{2/3} a^{2/3} x^{2/3} \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )}{a^{2/3}}\right )}{48 \sqrt [3]{x \left (a x+\sqrt {-b+a^2 x^2}\right )}} \]

input

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

output

(x^(1/3)*((-12*b*x^(2/3))/(a*x + Sqrt[-b + a^2*x^2]) + 18*x^(2/3)*(a*x + S 
qrt[-b + a^2*x^2]) - (14*2^(1/3)*Sqrt[3]*b^(2/3)*(a*x + Sqrt[-b + a^2*x^2] 
)^(1/3)*ArcTan[(1 + (2*2^(1/3)*a^(1/3)*x^(1/3)*(a*x + Sqrt[-b + a^2*x^2])^ 
(1/3))/b^(1/3))/Sqrt[3]])/a^(2/3) - (14*2^(1/3)*b^(2/3)*(a*x + Sqrt[-b + a 
^2*x^2])^(1/3)*Log[-b^(1/3) + 2^(1/3)*a^(1/3)*x^(1/3)*(a*x + Sqrt[-b + a^2 
*x^2])^(1/3)])/a^(2/3) + (7*2^(1/3)*b^(2/3)*(a*x + Sqrt[-b + a^2*x^2])^(1/ 
3)*Log[b^(2/3) + 2^(1/3)*a^(1/3)*b^(1/3)*x^(1/3)*(a*x + Sqrt[-b + a^2*x^2] 
)^(1/3) + 2^(2/3)*a^(2/3)*x^(2/3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)])/a^(2/ 
3)))/(48*(x*(a*x + Sqrt[-b + a^2*x^2]))^(1/3))
 

3.30.43.3 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.

\(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt [3]{x \sqrt {a^2 x^2-b}+a x^2}}dx\)

input

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

output

$Aborted
 

3.30.43.3.1 Defintions of rubi rules used

rule 7299

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

3.30.43.4 Maple [F]

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

input

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

output

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

3.30.43.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.30.43.6 Sympy [F]

\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x^2+x \sqrt {-b+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt [3]{x \left (a x + \sqrt {a^{2} x^{2} - b}\right )}}\, dx \]

input

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

output

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

3.30.43.7 Maxima [F]

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

input

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

output

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

3.30.43.8 Giac [F]

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

input

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

output

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

3.30.43.9 Mupad [F(-1)]

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

input

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

output

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