\(\int \frac {x^5 (7 b+10 a x^3)}{\sqrt [4]{b x^3+a x^6} (1+b x^7+a x^{10})} \, dx\) [2997]

Optimal result

Integrand size = 43, antiderivative size = 399 \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )+\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {-2 2^{5/6} x \sqrt [4]{b x^3+a x^6}+4 \sqrt [6]{2} x^2 \sqrt [4]{b x^3+a x^6}-\sqrt {2} x^3 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {b x^3+a x^6}+2\ 2^{2/3} x^3 \sqrt {b x^3+a x^6}-x^4 \sqrt {b x^3+a x^6}}\right ) \]

[Out]

-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)))

Rubi [F]

\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx \]

[In]

Int[(x^5*(7*b + 10*a*x^3))/((b*x^3 + a*x^6)^(1/4)*(1 + b*x^7 + a*x^10)),x]

[Out]

(28*b*x^(3/4)*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^20/((b + a*x^12)^(1/4)*(1 + b*x^28 + a*x^40)), x], x
, x^(1/4)])/(b*x^3 + a*x^6)^(1/4) + (40*a*x^(3/4)*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^32/((b + a*x^12)
^(1/4)*(1 + b*x^28 + a*x^40)), x], x, x^(1/4)])/(b*x^3 + a*x^6)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{b+a x^3}\right ) \int \frac {x^{17/4} \left (7 b+10 a x^3\right )}{\sqrt [4]{b+a x^3} \left (1+b x^7+a x^{10}\right )} \, dx}{\sqrt [4]{b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{20} \left (7 b+10 a x^{12}\right )}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )}+\frac {10 a x^{32}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}} \\ & = \frac {\left (40 a x^{3/4} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{32}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}+\frac {\left (28 b x^{3/4} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx \]

[In]

Integrate[(x^5*(7*b + 10*a*x^3))/((b*x^3 + a*x^6)^(1/4)*(1 + b*x^7 + a*x^10)),x]

[Out]

Integrate[(x^5*(7*b + 10*a*x^3))/((b*x^3 + a*x^6)^(1/4)*(1 + b*x^7 + a*x^10)), x]

Maple [F]

\[\int \frac {x^{5} \left (10 a \,x^{3}+7 b \right )}{\left (a \,x^{6}+b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{10}+b \,x^{7}+1\right )}d x\]

[In]

int(x^5*(10*a*x^3+7*b)/(a*x^6+b*x^3)^(1/4)/(a*x^10+b*x^7+1),x)

[Out]

int(x^5*(10*a*x^3+7*b)/(a*x^6+b*x^3)^(1/4)/(a*x^10+b*x^7+1),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^5*(10*a*x^3+7*b)/(a*x^6+b*x^3)^(1/4)/(a*x^10+b*x^7+1),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^{5} \cdot \left (10 a x^{3} + 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{3} + b\right )} \left (a x^{10} + b x^{7} + 1\right )}\, dx \]

[In]

integrate(x**5*(10*a*x**3+7*b)/(a*x**6+b*x**3)**(1/4)/(a*x**10+b*x**7+1),x)

[Out]

Integral(x**5*(10*a*x**3 + 7*b)/((x**3*(a*x**3 + b))**(1/4)*(a*x**10 + b*x**7 + 1)), x)

Maxima [F]

\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int { \frac {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^5*(10*a*x^3+7*b)/(a*x^6+b*x^3)^(1/4)/(a*x^10+b*x^7+1),x, algorithm="maxima")

[Out]

integrate((10*a*x^3 + 7*b)*x^5/((a*x^10 + b*x^7 + 1)*(a*x^6 + b*x^3)^(1/4)), x)

Giac [F]

\[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int { \frac {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^5*(10*a*x^3+7*b)/(a*x^6+b*x^3)^(1/4)/(a*x^10+b*x^7+1),x, algorithm="giac")

[Out]

integrate((10*a*x^3 + 7*b)*x^5/((a*x^10 + b*x^7 + 1)*(a*x^6 + b*x^3)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx=\int \frac {x^5\,\left (10\,a\,x^3+7\,b\right )}{{\left (a\,x^6+b\,x^3\right )}^{1/4}\,\left (a\,x^{10}+b\,x^7+1\right )} \,d x \]

[In]

int((x^5*(7*b + 10*a*x^3))/((a*x^6 + b*x^3)^(1/4)*(a*x^10 + b*x^7 + 1)),x)

[Out]

int((x^5*(7*b + 10*a*x^3))/((a*x^6 + b*x^3)^(1/4)*(a*x^10 + b*x^7 + 1)), x)