\(\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} (-b-a x^3+x^4)} \, dx\) [439]

Optimal result

Integrand size = 36, antiderivative size = 35 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-4*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]

Rubi steps \begin{align*} \text {integral}& = \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 (a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-x^4\right )} \, dx\right )-(4 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 = 0.80 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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