Integrand size = 45, antiderivative size = 177 \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\frac {\sqrt {-b^2 x+a^2 x^3}}{b^2-a^2 x^2}-\frac {\arctan \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{4 \sqrt {a} \sqrt {b}} \]
(a^2*x^3-b^2*x)^(1/2)/(-a^2*x^2+b^2)-1/4*arctan(2*a^(1/2)*b^(1/2)*(a^2*x^3 -b^2*x)^(1/2)/(a^2*x^2-2*a*b*x-b^2))/a^(1/2)/b^(1/2)-1/4*arctanh((-1/2*b^( 3/2)/a^(1/2)+a^(1/2)*b^(1/2)*x+1/2*a^(3/2)*x^2/b^(1/2))/(a^2*x^3-b^2*x)^(1 /2))/a^(1/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\frac {-4 \sqrt {a} \sqrt {b} x-(1-i) \sqrt {x} \sqrt {-b^2+a^2 x^2} \arctan \left (\frac {(1+i) \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(1+i) \sqrt {x} \sqrt {-b^2+a^2 x^2} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^2+a^2 x^2}}{\sqrt {a} \sqrt {b} \sqrt {x}}\right )}{4 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]
(-4*Sqrt[a]*Sqrt[b]*x - (1 - I)*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*ArcTan[((1 + I)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + (1 + I)*Sqrt[x]*Sqrt[- b^2 + a^2*x^2]*ArcTan[((1/2 + I/2)*Sqrt[-b^2 + a^2*x^2])/(Sqrt[a]*Sqrt[b]* Sqrt[x])])/(4*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])
Time = 1.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 25, 1388, 2035, 25, 7276, 2009}
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.
| \(\displaystyle \int \frac {a^4 x^4+b^4}{\sqrt {a^2 x^3-b^2 x} \left (a^4 x^4-b^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {b^4+a^4 x^4}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^4-a^4 x^4\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^4+a^4 x^4}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^4-a^4 x^4\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^4+a^4 x^4}{\sqrt {x} \left (-b^2-a^2 x^2\right ) \left (a^2 x^2-b^2\right )^{3/2}}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {b^4+a^4 x^4}{\left (a^2 x^2-b^2\right )^{3/2} \left (b^2+a^2 x^2\right )}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^4+a^4 x^4}{\left (a^2 x^2-b^2\right )^{3/2} \left (b^2+a^2 x^2\right )}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \left (\frac {2 b^4}{\left (a^2 x^2-b^2\right )^{3/2} \left (b^2+a^2 x^2\right )}-\frac {b^2}{\left (a^2 x^2-b^2\right )^{3/2}}+\frac {a^2 x^2}{\left (a^2 x^2-b^2\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{4 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{4 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b}}+\frac {\sqrt {x}}{2 \sqrt {a^2 x^2-b^2}}\right )}{\sqrt {a^2 x^3-b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*(Sqrt[x]/(2*Sqrt[-b^2 + a^2*x^2]) + ArcTa n[(Sqrt[2]*(-a^2)^(1/4)*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]]/(4*Sqrt[2]* (-a^2)^(1/4)*Sqrt[b]) + ArcTanh[(Sqrt[2]*(-a^2)^(1/4)*Sqrt[b]*Sqrt[x])/Sqr t[-b^2 + a^2*x^2]]/(4*Sqrt[2]*(-a^2)^(1/4)*Sqrt[b])))/Sqrt[-(b^2*x) + a^2* x^3]
3.24.9.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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 = 1.86 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.45
| method | result | size |
| default | \(\frac {\left (\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )}{2}-\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}-4 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{4 \sqrt {a^{2} x^{3}-b^{2} x}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(257\) |
| pseudoelliptic | \(\frac {\left (\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )}{2}-\arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}-4 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{4 \sqrt {a^{2} x^{3}-b^{2} x}\, \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(257\) |
| elliptic | \(-\frac {x}{\sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {b \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}-\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\) | \(322\) |
1/4/(a^2*x^3-b^2*x)^(1/2)/(a^2*b^2)^(1/4)*((arctan(((a^2*b^2)^(1/4)*x+(a^2 *x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x)+1/2*ln((a^2*x^2+2*(a^2*b^2)^(1/2)*x- b^2-2*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2))/(a^2*x^2+2*(a^2*b^2)^(1/4)*(a ^2*x^3-b^2*x)^(1/2)+2*(a^2*b^2)^(1/2)*x-b^2))-arctan(((a^2*b^2)^(1/4)*x-(a ^2*x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x))*(a^2*x^3-b^2*x)^(1/2)-4*(a^2*b^2) ^(1/4)*x)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 763, normalized size of antiderivative = 4.31 \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\frac {\left (\frac {1}{4}\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} - 6 \, a^{2} b^{2} x^{2} + b^{4} + 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} - 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{4}\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} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} - 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{4}\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} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} + 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{4}\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} - 6 \, a^{2} b^{2} x^{2} + b^{4} - 8 \, {\left (-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\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} + 4 \, {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}}}{a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - 8 \, \sqrt {a^{2} x^{3} - b^{2} x}}{8 \, {\left (a^{2} x^{2} - b^{2}\right )}} \]
1/8*((1/4)^(1/4)*(a^2*x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2 *b^2*x^2 + b^4 + 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(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) - 4 *(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - (1/4)^(1/4)*(a^2*x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6* a^2*b^2*x^2 + b^4 - 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^ (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) - 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + (1/4)^(1/4)*(-I*a^2*x^2 + I*b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4* x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(I*(1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4 ) + (1/4)^(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) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + (1/4)^(1/4)*(I*a^2*x^2 - I*b^2)*(-1/(a^2*b^2))^( 1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(-I*(1/4)^(1/4)*a^2*b^2*x*(-1/ (a^2*b^2))^(1/4) + (1/4)^(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) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^ 2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - 8*sqrt(a^2*x^3 - b^2*x))/(a^2*x^2 - b^2)
\[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\int \frac {a^{4} x^{4} + b^{4}}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )}\, dx \]
Integral((a**4*x**4 + b**4)/(sqrt(x*(a*x - b)*(a*x + b))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**2)), x)
\[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} + b^{4}}{{\left (a^{4} x^{4} - b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
\[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\int { \frac {a^{4} x^{4} + b^{4}}{{\left (a^{4} x^{4} - b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
Timed out. \[ \int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} \left (-b^4+a^4 x^4\right )} \, dx=\text {Hanged} \]