\(\int \frac {x^4 (-4 b+a x^3)}{\sqrt [4]{-b+a x^3} (-b^2+2 a b x^3-a^2 x^6+x^8)} \, dx\) [2044]

Optimal result

Integrand size = 52, antiderivative size = 146 \[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )-\frac {\arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )}{\sqrt {2}}+\text {arctanh}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right )}{\sqrt {2}} \]

[Out]

arctan((a*x^3-b)^(1/4)/x)-1/2*2^(1/2)*arctan((-1/2*2^(1/2)*x^2+1/2*(a*x^3-b)^(1/2)*2^(1/2))/x/(a*x^3-b)^(1/4))
+arctanh(x*(a*x^3-b)^(3/4)/(-a*x^3+b))+1/2*2^(1/2)*arctanh(2^(1/2)*x*(a*x^3-b)^(1/4)/(x^2+(a*x^3-b)^(1/2)))

Rubi [F]

\[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx \]

[In]

Int[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 b-a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {-4 b+a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx+\frac {1}{2} \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx \\ & = \frac {1}{2} \int \left (-\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )}-\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )} \, dx\right )+\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )-\frac {\arctan \left (\frac {-x^2+\sqrt {-b+a x^3}}{\sqrt {2} x \sqrt [4]{-b+a x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-b+a x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right )}{\sqrt {2}} \]

[In]

Integrate[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]

[Out]

ArcTan[(-b + a*x^3)^(1/4)/x] - ArcTan[(-x^2 + Sqrt[-b + a*x^3])/(Sqrt[2]*x*(-b + a*x^3)^(1/4))]/Sqrt[2] - ArcT
anh[x/(-b + a*x^3)^(1/4)] + ArcTanh[(Sqrt[2]*x*(-b + a*x^3)^(1/4))/(x^2 + Sqrt[-b + a*x^3])]/Sqrt[2]

Maple [F]

\[\int \frac {x^{4} \left (a \,x^{3}-4 b \right )}{\left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (-a^{2} x^{6}+x^{8}+2 a b \,x^{3}-b^{2}\right )}d x\]

[In]

int(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x)

[Out]

int(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate(x**4*(a*x**3-4*b)/(a*x**3-b)**(1/4)/(-a**2*x**6+x**8+2*a*b*x**3-b**2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\int { -\frac {{\left (a x^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x, algorithm="maxima")

[Out]

-integrate((a*x^3 - 4*b)*x^4/((a^2*x^6 - x^8 - 2*a*b*x^3 + b^2)*(a*x^3 - b)^(1/4)), x)

Giac [F]

\[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=\int { -\frac {{\left (a x^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^4*(a*x^3-4*b)/(a*x^3-b)^(1/4)/(-a^2*x^6+x^8+2*a*b*x^3-b^2),x, algorithm="giac")

[Out]

integrate(-(a*x^3 - 4*b)*x^4/((a^2*x^6 - x^8 - 2*a*b*x^3 + b^2)*(a*x^3 - b)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx=-\int -\frac {x^4\,\left (4\,b-a\,x^3\right )}{{\left (a\,x^3-b\right )}^{1/4}\,\left (a^2\,x^6-2\,a\,b\,x^3+b^2-x^8\right )} \,d x \]

[In]

int((x^4*(4*b - a*x^3))/((a*x^3 - b)^(1/4)*(b^2 - x^8 + a^2*x^6 - 2*a*b*x^3)),x)

[Out]

-int(-(x^4*(4*b - a*x^3))/((a*x^3 - b)^(1/4)*(b^2 - x^8 + a^2*x^6 - 2*a*b*x^3)), x)