Integrand size = 33, antiderivative size = 140 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}} \]
-1/2*arctan(2*a^(1/4)*b^(1/4)*(a*x^3-b*x)^(1/2)/(-b-2*a^(1/2)*b^(1/2)*x+a* x^2))/a^(1/4)/b^(1/4)-1/2*arctanh((-1/2*b^(3/4)/a^(1/4)+a^(1/4)*b^(1/4)*x+ 1/2*a^(3/4)*x^2/b^(1/4))/(a*x^3-b*x)^(1/2))/a^(1/4)/b^(1/4)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b x+a x^3} \left (i \arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+\arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \sqrt {-b+a x^2}} \]
((1/2 + I/2)*Sqrt[-(b*x) + a*x^3]*(I*ArcTan[((1 + I)*a^(1/4)*b^(1/4)*Sqrt[ x])/Sqrt[-b + a*x^2]] + ArcTan[((1/2 + I/2)*Sqrt[-b + a*x^2])/(a^(1/4)*b^( 1/4)*Sqrt[x])]))/(a^(1/4)*b^(1/4)*Sqrt[x]*Sqrt[-b + a*x^2])
Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2467, 368, 921}
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.
| \(\displaystyle \int \frac {a x^2-b}{\left (a x^2+b\right ) \sqrt {a x^3-b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2-b} \int \frac {\sqrt {a x^2-b}}{\sqrt {x} \left (a x^2+b\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \int \frac {\sqrt {a x^2-b}}{a x^2+b}d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 921 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a x^2-b} \left (-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}\right )}{\sqrt {a x^3-b x}}\) |
(2*Sqrt[x]*Sqrt[-b + a*x^2]*(-1/2*ArcTan[(a^(1/4)*Sqrt[x]*(Sqrt[b] - Sqrt[ a]*x))/(b^(1/4)*Sqrt[-b + a*x^2])]/(a^(1/4)*b^(1/4)) - ArcTanh[(a^(1/4)*Sq rt[x]*(Sqrt[b] + Sqrt[a]*x))/(b^(1/4)*Sqrt[-b + a*x^2])]/(2*a^(1/4)*b^(1/4 ))))/Sqrt[-(b*x) + a*x^3]
3.20.77.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*b, 4]}, Simp[(a/(2*c*q))*ArcTan[q*x*((a + q^2*x^2)/(a*Sqrt[a + b*x ^4]))], x] + Simp[(a/(2*c*q))*ArcTanh[q*x*((a - q^2*x^2)/(a*Sqrt[a + b*x^4] ))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]
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]
Time = 0.54 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )}{4 \left (a b \right )^{\frac {1}{4}}}\) | \(155\) |
| pseudoelliptic | \(\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )}{4 \left (a b \right )^{\frac {1}{4}}}\) | \(155\) |
| elliptic | \(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\) | \(380\) |
1/4/(a*b)^(1/4)*(ln((a*x^2+2*x*(a*b)^(1/2)-2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/ 2)-b)/(a*x^2+2*x*(a*b)^(1/2)+2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/2)-b))+2*arcta n((x*(a*b)^(1/4)+(x*(a*x^2-b))^(1/2))/(a*b)^(1/4)/x)-2*arctan((x*(a*b)^(1/ 4)-(x*(a*x^2-b))^(1/2))/(a*b)^(1/4)/x))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.15 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} + 8 \, \sqrt {a x^{3} - b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} - 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} - 8 \, \sqrt {a x^{3} - b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} - 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} - 8 \, \sqrt {a x^{3} - b x} {\left (i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (-i \, a^{2} b x^{2} + i \, a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} + 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} - 8 \, \sqrt {a x^{3} - b x} {\left (-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (i \, a^{2} b x^{2} - i \, a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} + 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) \]
1/4*(1/4)^(1/4)*(-1/(a*b))^(1/4)*log((a^2*x^4 - 6*a*b*x^2 + b^2 + 8*sqrt(a *x^3 - b*x)*((1/4)^(1/4)*a*b*x*(-1/(a*b))^(1/4) + (1/4)^(3/4)*(a^2*b*x^2 - a*b^2)*(-1/(a*b))^(3/4)) - 4*(a^2*b*x^3 - a*b^2*x)*sqrt(-1/(a*b)))/(a^2*x ^4 + 2*a*b*x^2 + b^2)) - 1/4*(1/4)^(1/4)*(-1/(a*b))^(1/4)*log((a^2*x^4 - 6 *a*b*x^2 + b^2 - 8*sqrt(a*x^3 - b*x)*((1/4)^(1/4)*a*b*x*(-1/(a*b))^(1/4) + (1/4)^(3/4)*(a^2*b*x^2 - a*b^2)*(-1/(a*b))^(3/4)) - 4*(a^2*b*x^3 - a*b^2* x)*sqrt(-1/(a*b)))/(a^2*x^4 + 2*a*b*x^2 + b^2)) - 1/4*I*(1/4)^(1/4)*(-1/(a *b))^(1/4)*log((a^2*x^4 - 6*a*b*x^2 + b^2 - 8*sqrt(a*x^3 - b*x)*(I*(1/4)^( 1/4)*a*b*x*(-1/(a*b))^(1/4) + (1/4)^(3/4)*(-I*a^2*b*x^2 + I*a*b^2)*(-1/(a* b))^(3/4)) + 4*(a^2*b*x^3 - a*b^2*x)*sqrt(-1/(a*b)))/(a^2*x^4 + 2*a*b*x^2 + b^2)) + 1/4*I*(1/4)^(1/4)*(-1/(a*b))^(1/4)*log((a^2*x^4 - 6*a*b*x^2 + b^ 2 - 8*sqrt(a*x^3 - b*x)*(-I*(1/4)^(1/4)*a*b*x*(-1/(a*b))^(1/4) + (1/4)^(3/ 4)*(I*a^2*b*x^2 - I*a*b^2)*(-1/(a*b))^(3/4)) + 4*(a^2*b*x^3 - a*b^2*x)*sqr t(-1/(a*b)))/(a^2*x^4 + 2*a*b*x^2 + b^2))
\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int \frac {a x^{2} - b}{\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} + b\right )}\, dx \]
\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} - b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} - b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
Timed out. \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\text {Hanged} \]