\(\int \frac {-2 b+a x^3}{(b+x^2+a x^3) \sqrt [4]{b x^2+a x^5}} \, dx\) [1659]

Optimal result

Integrand size = 37, antiderivative size = 112 \[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{b x^2+a x^5}}{-x^2+\sqrt {b x^2+a x^5}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b x^2+a x^5}}{\sqrt {2}}}{x \sqrt [4]{b x^2+a x^5}}\right ) \]

[Out]

-2^(1/2)*arctan(2^(1/2)*x*(a*x^5+b*x^2)^(1/4)/(-x^2+(a*x^5+b*x^2)^(1/2)))-2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2
*(a*x^5+b*x^2)^(1/2)*2^(1/2))/x/(a*x^5+b*x^2)^(1/4))

Rubi [F]

\[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx \]

[In]

Int[(-2*b + a*x^3)/((b + x^2 + a*x^3)*(b*x^2 + a*x^5)^(1/4)),x]

[Out]

(2*x*(1 + (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, -((a*x^3)/b)])/(b*x^2 + a*x^5)^(1/4) - (6*b*Sqrt[x
]*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][1/((b + a*x^6)^(1/4)*(b + x^4 + a*x^6)), x], x, Sqrt[x]])/(b*x^2 +
a*x^5)^(1/4) - (2*Sqrt[x]*(b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^4/((b + a*x^6)^(1/4)*(b + x^4 + a*x^6))
, x], x, Sqrt[x]])/(b*x^2 + a*x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^3}\right ) \int \frac {-2 b+a x^3}{\sqrt {x} \sqrt [4]{b+a x^3} \left (b+x^2+a x^3\right )} \, dx}{\sqrt [4]{b x^2+a x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {-2 b+a x^6}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{b+a x^6}}-\frac {3 b+x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {3 b+x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {3 b}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}+\frac {x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1+\frac {a x^3}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^6}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}} \\ & = \frac {2 x \sqrt [4]{1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-\frac {a x^3}{b}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03 \[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=-\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x^3} \left (\arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{b+a x^3}}{-x+\sqrt {b+a x^3}}\right )+\text {arctanh}\left (\frac {x+\sqrt {b+a x^3}}{\sqrt {2} \sqrt {x} \sqrt [4]{b+a x^3}}\right )\right )}{\sqrt [4]{x^2 \left (b+a x^3\right )}} \]

[In]

Integrate[(-2*b + a*x^3)/((b + x^2 + a*x^3)*(b*x^2 + a*x^5)^(1/4)),x]

[Out]

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

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{2} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{3}+b \right )}}{\left (x^{2} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{3}+b \right )}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{2}\)	\(136\)

[In]

int((a*x^3-2*b)/(a*x^3+x^2+b)/(a*x^5+b*x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\text {Timed out} \]

[In]

integrate((a*x^3-2*b)/(a*x^3+x^2+b)/(a*x^5+b*x^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\int \frac {a x^{3} - 2 b}{\sqrt [4]{x^{2} \left (a x^{3} + b\right )} \left (a x^{3} + b + x^{2}\right )}\, dx \]

[In]

integrate((a*x**3-2*b)/(a*x**3+x**2+b)/(a*x**5+b*x**2)**(1/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\int { \frac {a x^{3} - 2 \, b}{{\left (a x^{5} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{3} + x^{2} + b\right )}} \,d x } \]

[In]

integrate((a*x^3-2*b)/(a*x^3+x^2+b)/(a*x^5+b*x^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^3 - 2*b)/((a*x^5 + b*x^2)^(1/4)*(a*x^3 + x^2 + b)), x)

Giac [F]

\[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\int { \frac {a x^{3} - 2 \, b}{{\left (a x^{5} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{3} + x^{2} + b\right )}} \,d x } \]

[In]

integrate((a*x^3-2*b)/(a*x^3+x^2+b)/(a*x^5+b*x^2)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^3 - 2*b)/((a*x^5 + b*x^2)^(1/4)*(a*x^3 + x^2 + b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx=\int -\frac {2\,b-a\,x^3}{{\left (a\,x^5+b\,x^2\right )}^{1/4}\,\left (a\,x^3+x^2+b\right )} \,d x \]

[In]

int(-(2*b - a*x^3)/((a*x^5 + b*x^2)^(1/4)*(b + a*x^3 + x^2)),x)

[Out]

int(-(2*b - a*x^3)/((a*x^5 + b*x^2)^(1/4)*(b + a*x^3 + x^2)), x)