Integrand size = 44, antiderivative size = 167 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {b^2 x+a^2 x^3}}{3 \left (b^2+a^2 x^2\right )}-\frac {2 \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \] Output:
-2*(a^2*x^3+b^2*x)^(1/2)/(3*a^2*x^2+3*b^2)-2/9*arctan(3^(1/4)*a^(1/2)*b^(1 /2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))*3^(3/4)/a^(1/2)/b^(1/2)-2/9*arcta nh(3^(1/4)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))*3^(3/4)/a^ (1/2)/b^(1/2)
Time = 0.83 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {x} \left (3 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+3^{3/4} \sqrt {b^2+a^2 x^2} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \] Input:
Integrate[(-b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(b^6 + a^6*x^6)),x]
Output:
(-2*Sqrt[x]*(3*Sqrt[a]*Sqrt[b]*Sqrt[x] + 3^(3/4)*Sqrt[b^2 + a^2*x^2]*ArcTa n[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + 3^(3/4)*Sqrt[b^ 2 + a^2*x^2]*ArcTanh[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2] ]))/(9*Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])
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.
| \(\displaystyle \int \frac {a^6 x^6-b^6}{\sqrt {a^2 x^3+b^2 x} \left (a^6 x^6+b^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int -\frac {b^6-a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^6-a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^6-a^6 x^6}{\sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^6}{\sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^6}{\sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
Input:
Int[(-b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(b^6 + a^6*x^6)),x]
Output:
$Aborted
Time = 1.76 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(182\) |
| pseudoelliptic | \(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}-\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(182\) |
| elliptic | \(-\frac {2 x}{3 \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 i b^{3}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 i \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 b^{3}}{3 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{9 b \,a^{2}}\) | \(341\) |
Input:
int((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x,method=_RETURNVERB OSE)
Output:
1/9*(2*arctan(1/3*(x*(a^2*x^2+b^2))^(1/2)/x*3^(3/4)/(a^2*b^2)^(1/4))*(x*(a ^2*x^2+b^2))^(1/2)-ln((-x*3^(1/4)*(a^2*b^2)^(1/4)-(x*(a^2*x^2+b^2))^(1/2)) /(x*3^(1/4)*(a^2*b^2)^(1/4)-(x*(a^2*x^2+b^2))^(1/2)))*(x*(a^2*x^2+b^2))^(1 /2)-2*x*3^(1/4)*(a^2*b^2)^(1/4))/(x*(a^2*x^2+b^2))^(1/2)*3^(3/4)/(a^2*b^2) ^(1/4)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (131) = 262\).
Time = 0.18 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.80 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {12 \, \sqrt {a^{2} x^{3} + b^{2} x} a^{2} b^{2} - 2 \cdot 3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \arctan \left (-\frac {{\left (3 \cdot 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} a^{2} b^{2} x - 3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} {\left (a^{2} x^{2} + b^{2}\right )}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{6 \, {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )}}\right ) + 3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 2 \, \sqrt {3} {\left (a^{2} x^{3} + b^{2} x\right )} \sqrt {a^{2} b^{2}} + 2 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} x + 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )}\right )}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) - 3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 2 \, \sqrt {3} {\left (a^{2} x^{3} + b^{2} x\right )} \sqrt {a^{2} b^{2}} - 2 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (3^{\frac {3}{4}} \left (a^{2} b^{2}\right )^{\frac {3}{4}} x + 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )}\right )}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{18 \, {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )}} \] Input:
integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm=" fricas")
Output:
-1/18*(12*sqrt(a^2*x^3 + b^2*x)*a^2*b^2 - 2*3^(3/4)*(a^2*b^2)^(3/4)*(a^2*x ^2 + b^2)*arctan(-1/6*(3*3^(1/4)*(a^2*b^2)^(1/4)*a^2*b^2*x - 3^(3/4)*(a^2* b^2)^(3/4)*(a^2*x^2 + b^2))*sqrt(a^2*x^3 + b^2*x)/(a^4*b^2*x^3 + a^2*b^4*x )) + 3^(3/4)*(a^2*b^2)^(3/4)*(a^2*x^2 + b^2)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 + 2*sqrt(3)*(a^2*x^3 + b^2*x)*sqrt(a^2*b^2) + 2*sqrt(a^2*x^3 + b^2*x )*(3^(3/4)*(a^2*b^2)^(3/4)*x + 3^(1/4)*(a^2*b^2)^(1/4)*(a^2*x^2 + b^2)))/( a^4*x^4 - a^2*b^2*x^2 + b^4)) - 3^(3/4)*(a^2*b^2)^(3/4)*(a^2*x^2 + b^2)*lo g((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 + 2*sqrt(3)*(a^2*x^3 + b^2*x)*sqrt(a^2*b^ 2) - 2*sqrt(a^2*x^3 + b^2*x)*(3^(3/4)*(a^2*b^2)^(3/4)*x + 3^(1/4)*(a^2*b^2 )^(1/4)*(a^2*x^2 + b^2)))/(a^4*x^4 - a^2*b^2*x^2 + b^4)))/(a^4*b^2*x^2 + a ^2*b^4)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \] Input:
integrate((a**6*x**6-b**6)/(a**2*x**3+b**2*x)**(1/2)/(a**6*x**6+b**6),x)
Output:
Integral((a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)/(sqrt(x*(a**2*x**2 + b**2))*(a**2*x**2 + b**2)*(a**4*x**4 - a**2* b**2*x**2 + b**4)), x)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \] Input:
integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm=" maxima")
Output:
integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 + b^2*x)), x)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \] Input:
integrate((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm=" giac")
Output:
integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 + b^2*x)), x)
Time = 12.87 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.20 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2-6\,\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}+3^{3/4}\,a^2\,x^2+3\,3^{1/4}\,a\,b\,x}{a^2\,x^2-\sqrt {3}\,a\,b\,x+b^2}\right )}{9\,\sqrt {a}\,\sqrt {b}}-\frac {2\,\sqrt {a^2\,x^3+b^2\,x}}{3\,\left (a^2\,x^2+b^2\right )}+\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2+3^{3/4}\,a^2\,x^2-3\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+\sqrt {3}\,a\,b\,x+b^2}\right )\,1{}\mathrm {i}}{9\,\sqrt {a}\,\sqrt {b}} \] Input:
int(-(b^6 - a^6*x^6)/((b^6 + a^6*x^6)*(b^2*x + a^2*x^3)^(1/2)),x)
Output:
(3^(3/4)*log((3^(3/4)*b^2 - 6*a^(1/2)*b^(1/2)*(b^2*x + a^2*x^3)^(1/2) + 3^ (3/4)*a^2*x^2 + 3*3^(1/4)*a*b*x)/(b^2 + a^2*x^2 - 3^(1/2)*a*b*x)))/(9*a^(1 /2)*b^(1/2)) - (2*(b^2*x + a^2*x^3)^(1/2))/(3*(b^2 + a^2*x^2)) + (3^(3/4)* log((3^(3/4)*b^2 + a^(1/2)*b^(1/2)*(b^2*x + a^2*x^3)^(1/2)*6i + 3^(3/4)*a^ 2*x^2 - 3*3^(1/4)*a*b*x)/(b^2 + a^2*x^2 + 3^(1/2)*a*b*x))*1i)/(9*a^(1/2)*b ^(1/2))
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\left (\int \frac {\sqrt {a^{2} x^{2}+b^{2}}}{\sqrt {x}\, a^{8} x^{8}+\sqrt {x}\, a^{6} b^{2} x^{6}+\sqrt {x}\, a^{2} b^{6} x^{2}+\sqrt {x}\, b^{8}}d x \right ) b^{6}+\left (\int \frac {\sqrt {x}\, \sqrt {a^{2} x^{2}+b^{2}}\, x^{5}}{a^{8} x^{8}+a^{6} b^{2} x^{6}+a^{2} b^{6} x^{2}+b^{8}}d x \right ) a^{6} \] Input:
int((a^6*x^6-b^6)/(a^2*x^3+b^2*x)^(1/2)/(a^6*x^6+b^6),x)
Output:
- int(sqrt(a**2*x**2 + b**2)/(sqrt(x)*a**8*x**8 + sqrt(x)*a**6*b**2*x**6 + sqrt(x)*a**2*b**6*x**2 + sqrt(x)*b**8),x)*b**6 + int((sqrt(x)*sqrt(a**2* x**2 + b**2)*x**5)/(a**8*x**8 + a**6*b**2*x**6 + a**2*b**6*x**2 + b**8),x) *a**6