Integrand size = 39, antiderivative size = 52 \[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b x+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {x \left (-b x+a x^3\right )^{3/4}}{-b+a x^2}\right ) \]
\[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=\int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx \]
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.
\(\displaystyle \int \frac {5 a x^3-7 b x}{\sqrt [4]{a x^3-b x} \left (-a x^2+b+x^7\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (5 a x^2-7 b\right )}{\sqrt [4]{a x^3-b x} \left (-a x^2+b+x^7\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^2-b} \int -\frac {x^{3/4} \left (7 b-5 a x^2\right )}{\sqrt [4]{a x^2-b} \left (x^7-a x^2+b\right )}dx}{\sqrt [4]{a x^3-b x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^2-b} \int \frac {x^{3/4} \left (7 b-5 a x^2\right )}{\sqrt [4]{a x^2-b} \left (x^7-a x^2+b\right )}dx}{\sqrt [4]{a x^3-b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \int \frac {x^{3/2} \left (7 b-5 a x^2\right )}{\sqrt [4]{a x^2-b} \left (x^7-a x^2+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^3-b x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \int \left (\frac {5 a x^{7/2}}{\sqrt [4]{a x^2-b} \left (-x^7+a x^2-b\right )}+\frac {7 b x^{3/2}}{\sqrt [4]{a x^2-b} \left (x^7-a x^2+b\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^3-b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^2-b} \left (5 a \int \frac {x^{7/2}}{\sqrt [4]{a x^2-b} \left (-x^7+a x^2-b\right )}d\sqrt [4]{x}+7 b \int \frac {x^{3/2}}{\sqrt [4]{a x^2-b} \left (x^7-a x^2+b\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^3-b x}}\) |
3.7.59.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]
\[\int \frac {5 a \,x^{3}-7 b x}{\left (a \,x^{3}-b x \right )^{\frac {1}{4}} \left (x^{7}-a \,x^{2}+b \right )}d x\]
Timed out. \[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=\int \frac {x \left (5 a x^{2} - 7 b\right )}{\sqrt [4]{x \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{7}\right )}\, dx \]
\[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=\int { \frac {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=\int { \frac {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx=-\int \frac {7\,b\,x-5\,a\,x^3}{{\left (a\,x^3-b\,x\right )}^{1/4}\,\left (x^7-a\,x^2+b\right )} \,d x \]