Integrand size = 38, antiderivative size = 49 \[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b x^2+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {\left (b x^2+a x^3\right )^{3/4}}{b+a x}\right ) \]
Time = 10.78 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b x^2+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {\left (b x^2+a x^3\right )^{3/4}}{b+a x}\right ) \]
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.
\(\displaystyle \int \frac {x (5 a x+6 b)}{\sqrt [4]{a x^3+b x^2} \left (-a x-b+x^6\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x+b} \int -\frac {\sqrt {x} (6 b+5 a x)}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}dx}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x+b} \int \frac {\sqrt {x} (6 b+5 a x)}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}dx}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x+b} \int \frac {x (6 b+5 a x)}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}d\sqrt {x}}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x+b} \int \left (\frac {5 a x^2}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}+\frac {6 b x}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}\right )d\sqrt {x}}{\sqrt [4]{a x^3+b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x+b} \left (6 b \int \frac {x}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}d\sqrt {x}+5 a \int \frac {x^2}{\sqrt [4]{b+a x} \left (-x^6+a x+b\right )}d\sqrt {x}\right )}{\sqrt [4]{a x^3+b x^2}}\) |
3.7.30.3.1 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]
\[\int \frac {x \left (5 a x +6 b \right )}{\left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}} \left (x^{6}-a x -b \right )}d x\]
Timed out. \[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=\int \frac {x \left (5 a x + 6 b\right )}{\sqrt [4]{x^{2} \left (a x + b\right )} \left (- a x - b + x^{6}\right )}\, dx \]
\[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=\int { \frac {{\left (5 \, a x + 6 \, b\right )} x}{{\left (x^{6} - a x - b\right )} {\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=\int { \frac {{\left (5 \, a x + 6 \, b\right )} x}{{\left (x^{6} - a x - b\right )} {\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x (6 b+5 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^6\right )} \, dx=\int -\frac {x\,\left (6\,b+5\,a\,x\right )}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (-x^6+a\,x+b\right )} \,d x \]