\(\int \frac {3 b+a x^4}{(-b+x^3+a x^4) \sqrt [4]{-b x+a x^5}} \, dx\) [1576]

Optimal result

Integrand size = 38, antiderivative size = 108 \[ \int \frac {3 b+a x^4}{\left (-b+x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^5}}{-x^2+\sqrt {-b x+a x^5}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^5}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^5}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*x*(1 - (a*x^4)/b)^(1/4)*Hypergeometric2F1[3/16, 1/4, 19/16, (a*x^4)/b])/(3*(-(b*x) + a*x^5)^(1/4)) - (16*b*
x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^2/((b - x^12 - a*x^16)*(-b + a*x^16)^(1/4)), x], x, x^(1/
4)])/(-(b*x) + a*x^5)^(1/4) - (4*x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^16)^(1/4)*
(-b + x^12 + a*x^16)), x], x, x^(1/4)])/(-(b*x) + a*x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \int \frac {3 b+a x^4}{\sqrt [4]{x} \sqrt [4]{-b+a x^4} \left (-b+x^3+a x^4\right )} \, dx}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 b+a x^{16}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{-b+a x^{16}}}+\frac {x^2 \left (4 b-x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (4 b-x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 b x^2}{\left (b-x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}}-\frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1-\frac {a x^{16}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {4 x \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},\frac {a x^4}{b}\right )}{3 \sqrt [4]{-b x+a x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}-\frac {\left (16 b \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b-x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.87 \[ \int \frac {3 b+a x^4}{\left (-b+x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^5}}{-x^2+\sqrt {-b x+a x^5}}\right )+\text {arctanh}\left (\frac {x^2+\sqrt {-b x+a x^5}}{\sqrt {2} x \sqrt [4]{-b x+a x^5}}\right )\right ) \]

[In]

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

[Out]

-(Sqrt[2]*(ArcTan[(Sqrt[2]*x*(-(b*x) + a*x^5)^(1/4))/(-x^2 + Sqrt[-(b*x) + a*x^5])] + ArcTanh[(x^2 + Sqrt[-(b*
x) + a*x^5])/(Sqrt[2]*x*(-(b*x) + a*x^5)^(1/4))]))

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.26

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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