\(\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} (b+a x^3+x^4)} \, dx\) [1300]

Optimal result

Integrand size = 33, antiderivative size = 94 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{b+a x^3}}{x^2+\sqrt {b+a x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[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 \\ & = a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx+(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 = 0.94 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx=\sqrt {2} \left (-\arctan \left (\frac {-x^2+\sqrt {b+a x^3}}{\sqrt {2} x \sqrt [4]{b+a x^3}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{b+a x^3}}{x^2+\sqrt {b+a x^3}}\right )\right ) \]

[In]

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

[Out]

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

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)