\(\int \frac {(-3 b+a x^2) (b-a x^2+x^3)}{x^3 (-b+a x^2+x^3) \sqrt [4]{-b x+a x^3}} \, dx\) [1859]

Optimal result

Integrand size = 52, antiderivative size = 128 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-4*b*(1 - (a*x^2)/b)^(1/4)*Hypergeometric2F1[-9/8, 1/4, -1/8, (a*x^2)/b])/(3*x^2*(-(b*x) + a*x^3)^(1/4)) + (4
*a*(1 - (a*x^2)/b)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, (a*x^2)/b])/(-(b*x) + a*x^3)^(1/4) + (24*b*x^(1/4)*
(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^2/((-b + a*x^8)^(1/4)*(b - a*x^8 - x^12)), x], x, x^(1/4)])/(-(b*
x) + a*x^3)^(1/4) + (8*a*x^(1/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^10/((-b + a*x^8)^(1/4)*(-b + a*x
^8 + x^12)), x], x, x^(1/4)])/(-(b*x) + a*x^3)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^{13/4} \sqrt [4]{-b+a x^2} \left (-b+a x^2+x^3\right )} \, dx}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (-3 b+a x^8\right ) \left (b-a x^8+x^{12}\right )}{x^{10} \sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {3 b}{x^{10} \sqrt [4]{-b+a x^8}}-\frac {a}{x^2 \sqrt [4]{-b+a x^8}}+\frac {2 x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {3 b x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}+\frac {a x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ & = -\frac {4 b \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{8},\frac {1}{4},-\frac {1}{8},\frac {a x^2}{b}\right )}{3 x^2 \sqrt [4]{-b x+a x^3}}+\frac {4 a \sqrt [4]{1-\frac {a x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {1}{4},\frac {7}{8},\frac {a x^2}{b}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (24 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.78 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx=\frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b x+a x^3}}{\sqrt {2} x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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