\(\int \frac {b+2 a x^3}{(-b+x+a x^3) \sqrt [4]{-b x^3+a x^6}} \, dx\) [1722]

Optimal result

Integrand size = 37, antiderivative size = 116 \[ \int \frac {b+2 a x^3}{\left (-b+x+a x^3\right ) \sqrt [4]{-b x^3+a x^6}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-b x^3+a x^6}}{-x^2+\sqrt {-b x^3+a x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^3+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x^3+a x^6}}\right ) \]

[Out]

-2^(1/2)*arctan(2^(1/2)*x*(a*x^6-b*x^3)^(1/4)/(-x^2+(a*x^6-b*x^3)^(1/2)))-2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2
*(a*x^6-b*x^3)^(1/2)*2^(1/2))/x/(a*x^6-b*x^3)^(1/4))

Rubi [F]

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

[In]

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

[Out]

(8*x*(1 - (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/12, 1/4, 13/12, (a*x^3)/b])/(-(b*x^3) + a*x^6)^(1/4) - (12*b*x^
(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][1/((b - x^4 - a*x^12)*(-b + a*x^12)^(1/4)), x], x, x^(1/4)])/
(-(b*x^3) + a*x^6)^(1/4) - (8*x^(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^4/((-b + a*x^12)^(1/4)*(-b
+ x^4 + a*x^12)), x], x, x^(1/4)])/(-(b*x^3) + a*x^6)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{-b+a x^3}\right ) \int \frac {b+2 a x^3}{x^{3/4} \sqrt [4]{-b+a x^3} \left (-b+x+a x^3\right )} \, dx}{\sqrt [4]{-b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {b+2 a x^{12}}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {2}{\sqrt [4]{-b+a x^{12}}}+\frac {3 b-2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {3 b-2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \left (-\frac {3 b}{\left (b-x^4-a x^{12}\right ) \sqrt [4]{-b+a x^{12}}}-\frac {2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{1-\frac {a x^3}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^{12}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}} \\ & = \frac {8 x \sqrt [4]{1-\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{12},\frac {1}{4},\frac {13}{12},\frac {a x^3}{b}\right )}{\sqrt [4]{-b x^3+a x^6}}-\frac {\left (8 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}-\frac {\left (12 b x^{3/4} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (b-x^4-a x^{12}\right ) \sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{3} \left (a \,x^{3}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (a \,x^{3}-b \right )}}{\left (x^{3} \left (a \,x^{3}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (a \,x^{3}-b \right )}}\right )+2 \arctan \left (\frac {\left (x^{3} \left (a \,x^{3}-b \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{3} \left (a \,x^{3}-b \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{2}\)	\(148\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

Integral((2*a*x**3 + b)/((x**3*(a*x**3 - b))**(1/4)*(a*x**3 - b + x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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