\(\int \frac {(-4 b+a x^3) (b-a x^3+x^4)}{x^4 \sqrt [4]{-b+a x^3} (-b+a x^3+x^4)} \, dx\) [1813]

Optimal result

Integrand size = 51, antiderivative size = 123 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]

[Out]

4/3*(a*x^3-b)^(3/4)/x^3-2*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))+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]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 b}{x^4 \sqrt [4]{-b+a x^3}}-\frac {a}{x \sqrt [4]{-b+a x^3}}+\frac {2 \left (4 b-a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}\right ) \, dx \\ & = 2 \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-a \int \frac {1}{x \sqrt [4]{-b+a x^3}} \, dx+(4 b) \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx \\ & = 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-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+\frac {1}{3} (4 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x}} \, dx,x,x^3\right ) \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {\sqrt {2} a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\sqrt {2} a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 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 = 1.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b+a x^3}}{\sqrt {2} x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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