\(\int \frac {x^3 (5 b+8 a x^3)}{\sqrt [4]{b x+a x^4} (-2+b x^5+a x^8)} \, dx\) [733]

Optimal result

Integrand size = 41, antiderivative size = 56 \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=2^{3/4} \arctan \left (\frac {x \sqrt [4]{b x+a x^4}}{\sqrt [4]{2}}\right )-2^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{b x+a x^4}}{\sqrt [4]{2}}\right ) \]

[Out]

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))

Rubi [F]

\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx \]

[In]

Int[(x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(-2 + b*x^5 + a*x^8)),x]

[Out]

(20*b*x^(1/4)*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^14/((b + a*x^12)^(1/4)*(-2 + b*x^20 + a*x^32)), x],
x, x^(1/4)])/(b*x + a*x^4)^(1/4) + (32*a*x^(1/4)*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^26/((b + a*x^12)^
(1/4)*(-2 + b*x^20 + a*x^32)), x], x, x^(1/4)])/(b*x + a*x^4)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {x^{11/4} \left (5 b+8 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-2+b x^5+a x^8\right )} \, dx}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{14} \left (5 b+8 a x^{12}\right )}{\sqrt [4]{b+a x^{12}} \left (-2+b x^{20}+a x^{32}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {5 b x^{14}}{\sqrt [4]{b+a x^{12}} \left (-2+b x^{20}+a x^{32}\right )}+\frac {8 a x^{26}}{\sqrt [4]{b+a x^{12}} \left (-2+b x^{20}+a x^{32}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ & = \frac {\left (32 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{26}}{\sqrt [4]{b+a x^{12}} \left (-2+b x^{20}+a x^{32}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}}+\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{12}} \left (-2+b x^{20}+a x^{32}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx \]

[In]

Integrate[(x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(-2 + b*x^5 + a*x^8)),x]

[Out]

Integrate[(x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(-2 + b*x^5 + a*x^8)), x]

Maple [F]

\[\int \frac {x^{3} \left (8 a \,x^{3}+5 b \right )}{\left (a \,x^{4}+b x \right )^{\frac {1}{4}} \left (a \,x^{8}+b \,x^{5}-2\right )}d x\]

[In]

int(x^3*(8*a*x^3+5*b)/(a*x^4+b*x)^(1/4)/(a*x^8+b*x^5-2),x)

[Out]

int(x^3*(8*a*x^3+5*b)/(a*x^4+b*x)^(1/4)/(a*x^8+b*x^5-2),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^3*(8*a*x^3+5*b)/(a*x^4+b*x)^(1/4)/(a*x^8+b*x^5-2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^{3} \cdot \left (8 a x^{3} + 5 b\right )}{\sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{8} + b x^{5} - 2\right )}\, dx \]

[In]

integrate(x**3*(8*a*x**3+5*b)/(a*x**4+b*x)**(1/4)/(a*x**8+b*x**5-2),x)

[Out]

Integral(x**3*(8*a*x**3 + 5*b)/((x*(a*x**3 + b))**(1/4)*(a*x**8 + b*x**5 - 2)), x)

Maxima [F]

\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int { \frac {{\left (8 \, a x^{3} + 5 \, b\right )} x^{3}}{{\left (a x^{8} + b x^{5} - 2\right )} {\left (a x^{4} + b x\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^3*(8*a*x^3+5*b)/(a*x^4+b*x)^(1/4)/(a*x^8+b*x^5-2),x, algorithm="maxima")

[Out]

integrate((8*a*x^3 + 5*b)*x^3/((a*x^8 + b*x^5 - 2)*(a*x^4 + b*x)^(1/4)), x)

Giac [F]

\[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int { \frac {{\left (8 \, a x^{3} + 5 \, b\right )} x^{3}}{{\left (a x^{8} + b x^{5} - 2\right )} {\left (a x^{4} + b x\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^3*(8*a*x^3+5*b)/(a*x^4+b*x)^(1/4)/(a*x^8+b*x^5-2),x, algorithm="giac")

[Out]

integrate((8*a*x^3 + 5*b)*x^3/((a*x^8 + b*x^5 - 2)*(a*x^4 + b*x)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (5 b+8 a x^3\right )}{\sqrt [4]{b x+a x^4} \left (-2+b x^5+a x^8\right )} \, dx=\int \frac {x^3\,\left (8\,a\,x^3+5\,b\right )}{{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (a\,x^8+b\,x^5-2\right )} \,d x \]

[In]

int((x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(a*x^8 + b*x^5 - 2)),x)

[Out]

int((x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(a*x^8 + b*x^5 - 2)), x)