Integrand size = 33, antiderivative size = 110 \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b x^2+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{b x^2+a x^3}}{x^2+\sqrt {b x^2+a x^3}}\right ) \]
-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 = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x} \left (-\arctan \left (\frac {-x+\sqrt {b+a x}}{\sqrt {2} \sqrt {x} \sqrt [4]{b+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x}}{x+\sqrt {b+a x}}\right )\right )}{\sqrt [4]{x^2 (b+a x)}} \]
(Sqrt[2]*Sqrt[x]*(b + a*x)^(1/4)*(-ArcTan[(-x + Sqrt[b + a*x])/(Sqrt[2]*Sq rt[x]*(b + a*x)^(1/4))] + ArcTanh[(Sqrt[2]*Sqrt[x]*(b + a*x)^(1/4))/(x + S qrt[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 = 1.14 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.80, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, 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 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x+b} \int \frac {2 b+a x}{\sqrt {x} \sqrt [4]{b+a x} \left (x^2+a x+b\right )}dx}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x+b} \int \frac {2 b+a x}{\sqrt [4]{b+a x} \left (x^2+a x+b\right )}d\sqrt {x}}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 2256 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x+b} \int \left (\frac {a-\sqrt {a^2-4 b}}{\left (a+2 x-\sqrt {a^2-4 b}\right ) \sqrt [4]{b+a x}}+\frac {a+\sqrt {a^2-4 b}}{\left (a+2 x+\sqrt {a^2-4 b}\right ) \sqrt [4]{b+a x}}\right )d\sqrt {x}}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} \sqrt {x} \sqrt {-a \sqrt {a^2-4 b}-a^2+2 b}}-\frac {\sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} \sqrt {x} \sqrt {-a \sqrt {a^2-4 b}-a^2+2 b}}+\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \operatorname {EllipticPi}\left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} \sqrt {x} \sqrt {a \sqrt {a^2-4 b}-a^2+2 b}}-\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \operatorname {EllipticPi}\left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}},\arcsin \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} \sqrt {x} \sqrt {a \sqrt {a^2-4 b}-a^2+2 b}}\right )}{\sqrt [4]{a x^3+b x^2}}\) |
(2*Sqrt[x]*(b + a*x)^(1/4)*(((a + Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b) ]*EllipticPi[-((Sqrt[2]*Sqrt[b])/Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b]), Ar cSin[(b + a*x)^(1/4)/b^(1/4)], -1])/(Sqrt[2]*Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b]*Sqrt[x]) - ((a + Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*Ellipti cPi[(Sqrt[2]*Sqrt[b])/Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b], ArcSin[(b + a* x)^(1/4)/b^(1/4)], -1])/(Sqrt[2]*Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b]*Sqrt [x]) + ((a - Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*EllipticPi[-((Sqrt[ 2]*Sqrt[b])/Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]), ArcSin[(b + a*x)^(1/4)/ b^(1/4)], -1])/(Sqrt[2]*Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]*Sqrt[x]) - (( a - Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*EllipticPi[(Sqrt[2]*Sqrt[b]) /Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b], ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1 ])/(Sqrt[2]*Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]*Sqrt[x])))/(b*x^2 + a*x^3 )^(1/4)
3.17.10.3.1 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. \[ \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx=\int \frac {2\,b+a\,x}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (x^2+a\,x+b\right )} \,d x \]