\(\int \frac {-3 b+2 a x^5}{(2 b+x^3+2 a x^5) \sqrt [4]{b x+a x^6}} \, dx\) [1618]

Optimal result

Integrand size = 39, antiderivative size = 110 \[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{b x+a x^6}}{-x^2+\sqrt {2} \sqrt {b x+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{2^{3/4}}+\frac {\sqrt {b x+a x^6}}{\sqrt [4]{2}}}{x \sqrt [4]{b x+a x^6}}\right )}{\sqrt [4]{2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx \]

[In]

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

[Out]

(4*x*(1 + (a*x^5)/b)^(1/4)*Hypergeometric2F1[3/20, 1/4, 23/20, -((a*x^5)/b)])/(3*(b*x + a*x^6)^(1/4)) - (20*b*
x^(1/4)*(b + a*x^5)^(1/4)*Defer[Subst][Defer[Int][x^2/((b + a*x^20)^(1/4)*(2*b + x^12 + 2*a*x^20)), x], x, x^(
1/4)])/(b*x + a*x^6)^(1/4) - (4*x^(1/4)*(b + a*x^5)^(1/4)*Defer[Subst][Defer[Int][x^14/((b + a*x^20)^(1/4)*(2*
b + x^12 + 2*a*x^20)), x], x, x^(1/4)])/(b*x + a*x^6)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \int \frac {-3 b+2 a x^5}{\sqrt [4]{x} \sqrt [4]{b+a x^5} \left (2 b+x^3+2 a x^5\right )} \, dx}{\sqrt [4]{b x+a x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (-3 b+2 a x^{20}\right )}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{b+a x^{20}}}+\frac {x^2 \left (-5 b-x^{12}\right )}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (-5 b-x^{12}\right )}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \left (-\frac {5 b x^2}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )}-\frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+\frac {a x^5}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+\frac {a x^{20}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}} \\ & = \frac {4 x \sqrt [4]{1+\frac {a x^5}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-\frac {a x^5}{b}\right )}{3 \sqrt [4]{b x+a x^6}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}-\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx \]

[In]

Integrate[(-3*b + 2*a*x^5)/((2*b + x^3 + 2*a*x^5)*(b*x + a*x^6)^(1/4)),x]

[Out]

Integrate[(-3*b + 2*a*x^5)/((2*b + x^3 + 2*a*x^5)*(b*x + a*x^6)^(1/4)), x]

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25

method	result	size
pseudoelliptic	\(\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {-2 \,2^{\frac {1}{4}} {\left (x \left (a \,x^{5}+b \right )\right )}^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+2 \sqrt {x \left (a \,x^{5}+b \right )}}{2 \,2^{\frac {1}{4}} {\left (x \left (a \,x^{5}+b \right )\right )}^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+2 \sqrt {x \left (a \,x^{5}+b \right )}}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{5}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}+x}{x}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{5}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}-x}{x}\right )\right )}{4}\)	\(137\)

[In]

int((2*a*x^5-3*b)/(2*a*x^5+x^3+2*b)/(a*x^6+b*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/4*2^(3/4)*(ln((-2*2^(1/4)*(x*(a*x^5+b))^(1/4)*x+2^(1/2)*x^2+2*(x*(a*x^5+b))^(1/2))/(2*2^(1/4)*(x*(a*x^5+b))^
(1/4)*x+2^(1/2)*x^2+2*(x*(a*x^5+b))^(1/2)))+2*arctan(((x*(a*x^5+b))^(1/4)*2^(3/4)+x)/x)+2*arctan(((x*(a*x^5+b)
)^(1/4)*2^(3/4)-x)/x))

Fricas [F(-1)]

Timed out. \[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\text {Timed out} \]

[In]

integrate((2*a*x^5-3*b)/(2*a*x^5+x^3+2*b)/(a*x^6+b*x)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int \frac {2 a x^{5} - 3 b}{\sqrt [4]{x \left (a x^{5} + b\right )} \left (2 a x^{5} + 2 b + x^{3}\right )}\, dx \]

[In]

integrate((2*a*x**5-3*b)/(2*a*x**5+x**3+2*b)/(a*x**6+b*x)**(1/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int { \frac {2 \, a x^{5} - 3 \, b}{{\left (a x^{6} + b x\right )}^{\frac {1}{4}} {\left (2 \, a x^{5} + x^{3} + 2 \, b\right )}} \,d x } \]

[In]

integrate((2*a*x^5-3*b)/(2*a*x^5+x^3+2*b)/(a*x^6+b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*a*x^5 - 3*b)/((a*x^6 + b*x)^(1/4)*(2*a*x^5 + x^3 + 2*b)), x)

Giac [F]

\[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int { \frac {2 \, a x^{5} - 3 \, b}{{\left (a x^{6} + b x\right )}^{\frac {1}{4}} {\left (2 \, a x^{5} + x^{3} + 2 \, b\right )}} \,d x } \]

[In]

integrate((2*a*x^5-3*b)/(2*a*x^5+x^3+2*b)/(a*x^6+b*x)^(1/4),x, algorithm="giac")

[Out]

integrate((2*a*x^5 - 3*b)/((a*x^6 + b*x)^(1/4)*(2*a*x^5 + x^3 + 2*b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 a x^5\right ) \sqrt [4]{b x+a x^6}} \, dx=\int -\frac {3\,b-2\,a\,x^5}{{\left (a\,x^6+b\,x\right )}^{1/4}\,\left (2\,a\,x^5+x^3+2\,b\right )} \,d x \]

[In]

int(-(3*b - 2*a*x^5)/((b*x + a*x^6)^(1/4)*(2*b + 2*a*x^5 + x^3)),x)

[Out]

int(-(3*b - 2*a*x^5)/((b*x + a*x^6)^(1/4)*(2*b + 2*a*x^5 + x^3)), x)