Integrand size = 44, antiderivative size = 167 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {b^2 x+a^2 x^3}}{3 \left (b^2+a^2 x^2\right )}-\frac {2 \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]
-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 = 0.85 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {2 \sqrt {x} \left (3 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+3^{3/4} \sqrt {b^2+a^2 x^2} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]
(-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 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^6}{\sqrt {b^2+a^2 x^2} \left (b^6+a^6 x^6\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
3.23.43.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(178\) |
| pseudoelliptic | \(\frac {\left (2 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, 3^{\frac {3}{4}}}{3 x \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-\ln \left (\frac {x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{-x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}+\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}-2 x 3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{9 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(178\) |
| elliptic | \(-\frac {2 x}{3 \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right ) \left (a^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 i b^{3}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 i \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 b^{3}}{3 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{9 b \,a^{2}}\) | \(341\) |
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)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 742, normalized size of antiderivative = 4.44 \[ \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 {\left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} - i \, b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} + i \, b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (-i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2} x}}{6 \, {\left (a^{2} x^{2} + b^{2}\right )}} \]
-1/6*((1/3)^(1/4)*(a^2*x^2 + b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 5*a^2 *b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)) + 6*((1/3)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^2*x^2 + a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 + b^2*x))/(a^4*x^4 - a^2*b^2*x ^2 + b^4)) - (1/3)^(1/4)*(a^2*x^2 + b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2* b^2)) - 6*((1/3)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3)^(3/4)*(a^4*b^ 2*x^2 + a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 + b^2*x))/(a^4*x^4 - a^ 2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(-I*a^2*x^2 - I*b^2)*(1/(a^2*b^2))^(1/4)*l og((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 + a^2*b^4*x)* sqrt(1/(a^2*b^2)) - 6*(I*(1/3)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + (1/3) ^(3/4)*(-I*a^4*b^2*x^2 - I*a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 + b^ 2*x))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(I*a^2*x^2 + I*b^2)*(1/ (a^2*b^2))^(1/4)*log((a^4*x^4 + 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2 *x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)) - 6*(-I*(1/3)^(1/4)*a^2*b^2*x*(1/(a^2* b^2))^(1/4) + (1/3)^(3/4)*(I*a^4*b^2*x^2 + I*a^2*b^4)*(1/(a^2*b^2))^(3/4)) *sqrt(a^2*x^3 + b^2*x))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) + 4*sqrt(a^2*x^3 + b^2*x))/(a^2*x^2 + b^2)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]
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)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
Time = 10.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.20 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2-6\,\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}+3^{3/4}\,a^2\,x^2+3\,3^{1/4}\,a\,b\,x}{a^2\,x^2-\sqrt {3}\,a\,b\,x+b^2}\right )}{9\,\sqrt {a}\,\sqrt {b}}-\frac {2\,\sqrt {a^2\,x^3+b^2\,x}}{3\,\left (a^2\,x^2+b^2\right )}+\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2+3^{3/4}\,a^2\,x^2-3\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+\sqrt {3}\,a\,b\,x+b^2}\right )\,1{}\mathrm {i}}{9\,\sqrt {a}\,\sqrt {b}} \]
(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))