Integrand size = 43, antiderivative size = 399 \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )+\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {-2 2^{5/6} x \sqrt [4]{b x^3+a x^6}+4 \sqrt [6]{2} x^2 \sqrt [4]{b x^3+a x^6}-\sqrt {2} x^3 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {b x^3+a x^6}+2\ 2^{2/3} x^3 \sqrt {b x^3+a x^6}-x^4 \sqrt {b x^3+a x^6}}\right ) \]
-2^(1/2)*arctan((-2^(2/3)*x*(a*x^6+b*x^3)^(1/4)+x^2*(a*x^6+b*x^3)^(1/4))/( 2*2^(1/6)-x*2^(1/2)+2^(2/3)*x*(a*x^6+b*x^3)^(1/4)-x^2*(a*x^6+b*x^3)^(1/4)) )+2^(1/2)*arctan((-2^(2/3)*x*(a*x^6+b*x^3)^(1/4)+x^2*(a*x^6+b*x^3)^(1/4))/ (-2*2^(1/6)+x*2^(1/2)+2^(2/3)*x*(a*x^6+b*x^3)^(1/4)-x^2*(a*x^6+b*x^3)^(1/4 )))-2^(1/2)*arctanh((-2*2^(5/6)*x*(a*x^6+b*x^3)^(1/4)+4*2^(1/6)*x^2*(a*x^6 +b*x^3)^(1/4)-2^(1/2)*(a*x^6+b*x^3)^(1/4)*x^3)/(-2*2^(1/3)+2*2^(2/3)*x-x^2 -2*2^(1/3)*x^2*(a*x^6+b*x^3)^(1/2)+2*2^(2/3)*x^3*(a*x^6+b*x^3)^(1/2)-x^4*( a*x^6+b*x^3)^(1/2)))
\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx \]
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.
\(\displaystyle \int \frac {x^5 \left (10 a x^3+7 b\right )}{\sqrt [4]{a x^6+b x^3} \left (a x^{10}+b x^7+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a x^3+b} \int \frac {x^{17/4} \left (10 a x^3+7 b\right )}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}dx}{\sqrt [4]{a x^6+b x^3}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{a x^3+b} \int \frac {x^5 \left (10 a x^3+7 b\right )}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^6+b x^3}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{a x^3+b} \int \left (\frac {10 a x^8}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}+\frac {7 b x^5}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^6+b x^3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 x^{3/4} \sqrt [4]{a x^3+b} \left (10 a \int \frac {x^8}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}d\sqrt [4]{x}+7 b \int \frac {x^5}{\sqrt [4]{a x^3+b} \left (a x^{10}+b x^7+1\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^6+b x^3}}\) |
3.30.97.3.1 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]
\[\int \frac {x^{5} \left (10 a \,x^{3}+7 b \right )}{\left (a \,x^{6}+b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{10}+b \,x^{7}+1\right )}d x\]
Timed out. \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^{5} \cdot \left (10 a x^{3} + 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{3} + b\right )} \left (a x^{10} + b x^{7} + 1\right )}\, dx \]
\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int { \frac {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int { \frac {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^5\,\left (10\,a\,x^3+7\,b\right )}{{\left (a\,x^6+b\,x^3\right )}^{1/4}\,\left (a\,x^{10}+b\,x^7+1\right )} \,d x \]