Integrand size = 41, antiderivative size = 56 \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=2^{3/4} \arctan \left (\frac {x \sqrt [4]{b x+a x^4}}{\sqrt [4]{2}}\right )-2^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{b x+a x^4}}{\sqrt [4]{2}}\right ) \]
2^(3/4)*arctan(1/2*x*(a*x^4+b*x)^(1/4)*2^(3/4))-2^(3/4)*arctanh(1/2*x*(a*x ^4+b*x)^(1/4)*2^(3/4))
\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx \]
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.
\(\displaystyle \int \frac {x^3 \left (8 a x^3+5 b\right )}{\sqrt [4]{a x^4+b x} \left (a x^8+b x^5-2\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int -\frac {x^{11/4} \left (8 a x^3+5 b\right )}{\sqrt [4]{a x^3+b} \left (-a x^8-b x^5+2\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {x^{11/4} \left (8 a x^3+5 b\right )}{\sqrt [4]{a x^3+b} \left (-a x^8-b x^5+2\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {x^{7/2} \left (8 a x^3+5 b\right )}{\sqrt [4]{a x^3+b} \left (-a x^8-b x^5+2\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \left (-\frac {8 a x^{13/2}}{\sqrt [4]{a x^3+b} \left (a x^8+b x^5-2\right )}-\frac {5 b x^{7/2}}{\sqrt [4]{a x^3+b} \left (a x^8+b x^5-2\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (-5 b \int \frac {x^{7/2}}{\sqrt [4]{a x^3+b} \left (a x^8+b x^5-2\right )}d\sqrt [4]{x}-8 a \int \frac {x^{13/2}}{\sqrt [4]{a x^3+b} \left (a x^8+b x^5-2\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^4+b x}}\) |
3.8.33.3.1 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]
\[\int \frac {x^{3} \left (8 a \,x^{3}+5 b \right )}{\left (a \,x^{4}+b x \right )^{\frac {1}{4}} \left (a \,x^{8}+b \,x^{5}-2\right )}d x\]
Timed out. \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^{3} \cdot \left (8 a x^{3} + 5 b\right )}{\sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{8} + b x^{5} - 2\right )}\, dx \]
\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int { \frac {{\left (8 \, a x^{3} + 5 \, b\right )} x^{3}}{{\left (a x^{8} + b x^{5} - 2\right )} {\left (a x^{4} + b x\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int { \frac {{\left (8 \, a x^{3} + 5 \, b\right )} x^{3}}{{\left (a x^{8} + b x^{5} - 2\right )} {\left (a x^{4} + b x\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^3\,\left (8\,a\,x^3+5\,b\right )}{{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (a\,x^8+b\,x^5-2\right )} \,d x \]