\(\int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx\) [2913]

Optimal result

Integrand size = 48, antiderivative size = 328 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=-a^2 x+a \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}+\frac {b^4}{64 a^3 \left (-a x+\sqrt {b+a^2 x^2}\right )^4}-\frac {\left (-a x+\sqrt {b+a^2 x^2}\right )^4}{64 a^3}+\frac {b^2 \log \left (-a x+\sqrt {b+a^2 x^2}\right )}{8 a^3}+\frac {2 \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {a^4 b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a^6 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ]}{a^3} \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.98 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=-a^2 x+a \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}+\frac {b^4}{64 a^3 \left (-a x+\sqrt {b+a^2 x^2}\right )^4}-\frac {\left (-a x+\sqrt {b+a^2 x^2}\right )^4}{64 a^3}+\frac {b^2 \log \left (-a x+\sqrt {b+a^2 x^2}\right )}{8 a^3}+2 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ] \] Input:

Integrate[(x^4*Sqrt[b + a^2*x^2])/(x^2 - Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]
 

Output:

-(a^2*x) + a*Sqrt[b + a^2*x^2] + b^2/(6*a*(a*x - Sqrt[b + a^2*x^2])^(3/2)) 
 - (b*Sqrt[a*x - Sqrt[b + a^2*x^2]])/a - (a*x - Sqrt[b + a^2*x^2])^(5/2)/( 
10*a) + b^4/(64*a^3*(-(a*x) + Sqrt[b + a^2*x^2])^4) - (-(a*x) + Sqrt[b + a 
^2*x^2])^4/(64*a^3) + (b^2*Log[-(a*x) + Sqrt[b + a^2*x^2]])/(8*a^3) + 2*a* 
RootSum[b^2 - 2*b*#1^4 - 4*a^2*#1^5 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + 
a^2*x^2]] - #1]*#1^2 + a^2*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^3)/( 
2*b + 5*a^2*#1 - 2*#1^4) & ]
 

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.

\(\displaystyle \int \frac {x^4 \sqrt {a^2 x^2+b}}{x^2-\sqrt {a x-\sqrt {a^2 x^2+b}}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x^2 \sqrt {a^2 x^2+b}+\sqrt {a^2 x^2+b} \sqrt {a x-\sqrt {a^2 x^2+b}}-\frac {a x^5 \sqrt {a^2 x^2+b} \sqrt {a x-\sqrt {a^2 x^2+b}}}{2 a x^5+b-x^8}+\frac {x^2 \sqrt {a^2 x^2+b} \left (a x^5+b\right )}{-2 a x^5-b+x^8}-\frac {b \sqrt {a^2 x^2+b} \sqrt {a x-\sqrt {a^2 x^2+b}}}{2 a x^5+b-x^8}+\frac {x^6 \left (a^2 x^2+b\right )}{2 a x^5+b-x^8}+\frac {x^4 \left (a^2 x^2+b\right ) \sqrt {a x-\sqrt {a^2 x^2+b}}}{2 a x^5+b-x^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a^3 \int \frac {x^5}{-x^8+2 a x^5+b}dx+a^2 b \int \frac {1}{-x^8+2 a x^5+b}dx-a \int \frac {x^5 \sqrt {a^2 x^2+b} \sqrt {a x-\sqrt {a^2 x^2+b}}}{-x^8+2 a x^5+b}dx-b \int \frac {x^2 \sqrt {a^2 x^2+b}}{-x^8+2 a x^5+b}dx-b \int \frac {\sqrt {a^2 x^2+b} \sqrt {a x-\sqrt {a^2 x^2+b}}}{-x^8+2 a x^5+b}dx-a \int \frac {x^7 \sqrt {a^2 x^2+b}}{-x^8+2 a x^5+b}dx+a^2 \int \frac {x^6 \sqrt {a x-\sqrt {a^2 x^2+b}}}{-x^8+2 a x^5+b}dx+b \int \frac {x^4 \sqrt {a x-\sqrt {a^2 x^2+b}}}{-x^8+2 a x^5+b}dx+b \int \frac {x^6}{-x^8+2 a x^5+b}dx+\frac {b^2}{6 a \left (a x-\sqrt {a^2 x^2+b}\right )^{3/2}}-\frac {\left (a x-\sqrt {a^2 x^2+b}\right )^{5/2}}{10 a}-\frac {b \sqrt {a x-\sqrt {a^2 x^2+b}}}{a}+\frac {b x \sqrt {a^2 x^2+b}}{8 a^2}+\frac {1}{4} x^3 \sqrt {a^2 x^2+b}-a^2 x-\frac {b^2 \text {arctanh}\left (\frac {a x}{\sqrt {a^2 x^2+b}}\right )}{8 a^3}\)

Input:

Int[(x^4*Sqrt[b + a^2*x^2])/(x^2 - Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13

\[\int \frac {x^{4} \sqrt {a^{2} x^{2}+b}}{x^{2}-\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}d x\]

Input:

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)
 

Output:

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\text {Timed out} \] Input:

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int \frac {x^{4} \sqrt {a^{2} x^{2} + b}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}}\, dx \] Input:

integrate(x**4*(a**2*x**2+b)**(1/2)/(x**2-(a*x-(a**2*x**2+b)**(1/2))**(1/2 
)),x)
 

Output:

Integral(x**4*sqrt(a**2*x**2 + b)/(x**2 - sqrt(a*x - sqrt(a**2*x**2 + b))) 
, x)
 

Maxima [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b} x^{4}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}} \,d x } \] Input:

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, alg 
orithm="maxima")
 

Output:

integrate(sqrt(a^2*x^2 + b)*x^4/(x^2 - sqrt(a*x - sqrt(a^2*x^2 + b))), x)
 

Giac [N/A]

Not integrable

Time = 44.64 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b} x^{4}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}} \,d x } \] Input:

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, alg 
orithm="giac")
 

Output:

integrate(sqrt(a^2*x^2 + b)*x^4/(x^2 - sqrt(a*x - sqrt(a^2*x^2 + b))), x)
 

Mupad [N/A]

Not integrable

Time = 9.73 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.14 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=-\int \frac {x^4\,\sqrt {a^2\,x^2+b}}{\sqrt {a\,x-\sqrt {a^2\,x^2+b}}-x^2} \,d x \] Input:

int(-(x^4*(b + a^2*x^2)^(1/2))/((a*x - (b + a^2*x^2)^(1/2))^(1/2) - x^2),x 
)
 

Output:

-int((x^4*(b + a^2*x^2)^(1/2))/((a*x - (b + a^2*x^2)^(1/2))^(1/2) - x^2), 
x)
 

Reduce [N/A]

Not integrable

Time = 200.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int \frac {x^{4} \sqrt {a^{2} x^{2}+b}}{x^{2}-\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}d x \] Input:

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)
 

Output:

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)