\(\int \frac {4 b+x^3}{(b+x^3) \sqrt [4]{-b-x^3+a x^4}} \, dx\) [884]

Optimal result

Integrand size = 33, antiderivative size = 67 \[ \int \frac {4 b+x^3}{\left (b+x^3\right ) \sqrt [4]{-b-x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b-x^3+a x^4}}\right )}{\sqrt [4]{a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b-x^3+a x^4}}\right )}{\sqrt [4]{a}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {4 b+x^3}{\left (b+x^3\right ) \sqrt [4]{-b-x^3+a x^4}} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+x^3 (-1+a x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+x^3 (-1+a x)}}\right )\right )}{\sqrt [4]{a}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\left (a \,x^{4}-x^{3}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}-x^{3}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}-x^{3}-b \right )^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}\)	\(85\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.99 \[ \int \frac {4 b+x^3}{\left (b+x^3\right ) \sqrt [4]{-b-x^3+a x^4}} \, dx=\frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} - x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} - x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} - x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} - x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{a^{\frac {1}{4}}} \]

[In]

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

[Out]

log((a^(1/4)*x + (a*x^4 - x^3 - b)^(1/4))/x)/a^(1/4) - log(-(a^(1/4)*x - (a*x^4 - x^3 - b)^(1/4))/x)/a^(1/4) -
I*log((I*a^(1/4)*x + (a*x^4 - x^3 - b)^(1/4))/x)/a^(1/4) + I*log((-I*a^(1/4)*x + (a*x^4 - x^3 - b)^(1/4))/x)/
a^(1/4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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