Integrand size = 36, antiderivative size = 114 \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^3}}{x^2+\sqrt {-b x^2+a x^3}}\right ) \]
-2^(1/2)*arctan((-1/2*2^(1/2)*x^2+1/2*(a*x^3-b*x^2)^(1/2)*2^(1/2))/x/(a*x^ 3-b*x^2)^(1/4))+2^(1/2)*arctanh(2^(1/2)*x*(a*x^3-b*x^2)^(1/4)/(x^2+(a*x^3- b*x^2)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x} \left (-\arctan \left (\frac {-x+\sqrt {-b+a x}}{\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}{x+\sqrt {-b+a x}}\right )\right )}{\sqrt [4]{x^2 (-b+a x)}} \]
(Sqrt[2]*Sqrt[x]*(-b + a*x)^(1/4)*(-ArcTan[(-x + Sqrt[-b + a*x])/(Sqrt[2]* Sqrt[x]*(-b + a*x)^(1/4))] + ArcTanh[(Sqrt[2]*Sqrt[x]*(-b + a*x)^(1/4))/(x + Sqrt[-b + a*x])]))/(x^2*(-b + a*x))^(1/4)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 10.62 (sec) , antiderivative size = 2392, normalized size of antiderivative = 20.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2467, 2035, 2256, 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.
| \(\displaystyle \int \frac {a x-2 b}{\left (a x-b+x^2\right ) \sqrt [4]{a x^3-b x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x-b} \int \frac {2 b-a x}{\sqrt {x} \sqrt [4]{a x-b} \left (-x^2-a x+b\right )}dx}{\sqrt [4]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x-b} \int \frac {2 b-a x}{\sqrt [4]{a x-b} \left (-x^2-a x+b\right )}d\sqrt {x}}{\sqrt [4]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x-b} \int \left (\frac {-a-\sqrt {a^2+4 b}}{\left (-a-2 x-\sqrt {a^2+4 b}\right ) \sqrt [4]{a x-b}}+\frac {\sqrt {a^2+4 b}-a}{\left (-a-2 x+\sqrt {a^2+4 b}\right ) \sqrt [4]{a x-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x-b} \left (-\frac {\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}+\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {x}}+\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \left (\sqrt {2}-\frac {\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}}{\sqrt {b}}\right )^2}{4 \sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt {x}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \left (\sqrt {2}-\frac {\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}}{\sqrt {b}}\right )^2}{4 \sqrt {2} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {x}}\right )}{\sqrt [4]{a x^3-b x^2}}\) |
(2*Sqrt[x]*(-b + a*x)^(1/4)*(-1/2*(Sqrt[b]*Sqrt[a + Sqrt[a^2 + 4*b]]*Sqrt[ (a*x)/b]*ArcTan[(Sqrt[a]*Sqrt[a + Sqrt[a^2 + 4*b]]*(-b + a*x)^(1/4))/(2^(1 /4)*Sqrt[b]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/(2^(1/ 4)*Sqrt[a]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x]) - (Sqrt[b]*Sqrt [a - Sqrt[a^2 + 4*b]]*Sqrt[(a*x)/b]*ArcTan[(Sqrt[a]*Sqrt[a - Sqrt[a^2 + 4* b]]*(-b + a*x)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1 /4)*Sqrt[(a*x)/b])])/(2*2^(1/4)*Sqrt[a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^( 1/4)*Sqrt[x]) + (Sqrt[b]*Sqrt[a + Sqrt[a^2 + 4*b]]*Sqrt[(a*x)/b]*ArcTanh[( Sqrt[a]*Sqrt[a + Sqrt[a^2 + 4*b]]*(-b + a*x)^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x)/b])])/(2*2^(1/4)*Sqrt[a]*(-a^ 2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x]) + (Sqrt[b]*Sqrt[a - Sqrt[a^2 + 4*b]]*Sqrt[(a*x)/b]*ArcTanh[(Sqrt[a]*Sqrt[a - Sqrt[a^2 + 4*b]]*(-b + a*x) ^(1/4))/(2^(1/4)*Sqrt[b]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[(a*x) /b])])/(2*2^(1/4)*Sqrt[a]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*Sqrt[x]) - ((a + Sqrt[a^2 + 4*b])*(2*Sqrt[b] - Sqrt[2]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a* x])*EllipticF[2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(4*b^(1/4)*(a^2 + 4*b + a*Sqrt[a^2 + 4*b])*Sqrt[x]) - ((a + Sqrt[a^2 + 4*b])*(2*Sqrt[b] + Sq rt[2]*Sqrt[-a^2 - 2*b - a*Sqrt[a^2 + 4*b]])*Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticF[2*ArcTan[(-b + a*x)^(1/...
3.17.98.3.1 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[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
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]
\[\int \frac {a x -2 b}{\left (a x +x^{2}-b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}d x\]
Timed out. \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int \frac {a x - 2 b}{\sqrt [4]{x^{2} \left (a x - b\right )} \left (a x - b + x^{2}\right )}\, dx \]
\[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}} \,d x } \]
\[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}} \,d x } \]
Timed out. \[ \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx=\int -\frac {2\,b-a\,x}{{\left (a\,x^3-b\,x^2\right )}^{1/4}\,\left (x^2+a\,x-b\right )} \,d x \]