\(\int \frac {(2 b+a x^6) (-b-c x^4+a x^6)}{x^2 (-b+a x^6)^{3/4} (-b+c x^4+a x^6)} \, dx\) [2092]

Optimal result

Integrand size = 57, antiderivative size = 151 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-c x^4+a x^6\right )}{x^2 \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\frac {2 \sqrt [4]{-b+a x^6}}{x}+\sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}{-\sqrt {c} x^2+\sqrt {-b+a x^6}}\right )-\sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-2*c*Sqrt[(a*x^6)/(Sqrt[b] + Sqrt[-b + a*x^6])^2]*(Sqrt[b] + Sqrt[-b + a*x^6])*EllipticF[2*ArcTan[(-b + a*x^6
)^(1/4)/b^(1/4)], 1/2])/(3*a*b^(1/4)*x^3) - (2*b*(1 - (a*x^6)/b)^(3/4)*Hypergeometric2F1[-1/6, 3/4, 5/6, (a*x^
6)/b])/(x*(-b + a*x^6)^(3/4)) + (2*c^2*x*(1 - (a*x^6)/b)^(3/4)*Hypergeometric2F1[1/6, 3/4, 7/6, (a*x^6)/b])/(a
*(-b + a*x^6)^(3/4)) + (a*x^5*(1 - (a*x^6)/b)^(3/4)*Hypergeometric2F1[3/4, 5/6, 11/6, (a*x^6)/b])/(5*(-b + a*x
^6)^(3/4)) - (2*b*c^2*Defer[Int][1/((b - c*x^4 - a*x^6)*(-b + a*x^6)^(3/4)), x])/a - 6*b*c*Defer[Int][x^2/((-b
+ a*x^6)^(3/4)*(-b + c*x^4 + a*x^6)), x] - (2*c^3*Defer[Int][x^4/((-b + a*x^6)^(3/4)*(-b + c*x^4 + a*x^6)), x
])/a

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c^2}{a \left (-b+a x^6\right )^{3/4}}+\frac {2 b}{x^2 \left (-b+a x^6\right )^{3/4}}-\frac {2 c x^2}{\left (-b+a x^6\right )^{3/4}}+\frac {a x^4}{\left (-b+a x^6\right )^{3/4}}-\frac {2 c \left (-b c+3 a b x^2+c^2 x^4\right )}{a \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}\right ) \, dx \\ & = a \int \frac {x^4}{\left (-b+a x^6\right )^{3/4}} \, dx+(2 b) \int \frac {1}{x^2 \left (-b+a x^6\right )^{3/4}} \, dx-(2 c) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4}} \, dx-\frac {(2 c) \int \frac {-b c+3 a b x^2+c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a}+\frac {\left (2 c^2\right ) \int \frac {1}{\left (-b+a x^6\right )^{3/4}} \, dx}{a} \\ & = -\left (\frac {1}{3} (2 c) \text {Subst}\left (\int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx,x,x^3\right )\right )-\frac {(2 c) \int \left (\frac {b c}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}+\frac {3 a b x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}+\frac {c^2 x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )}\right ) \, dx}{a}+\frac {\left (a \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {x^4}{\left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{\left (-b+a x^6\right )^{3/4}}+\frac {\left (2 b \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{\left (-b+a x^6\right )^{3/4}}+\frac {\left (2 c^2 \left (1-\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {a x^6}{b}\right )^{3/4}} \, dx}{a \left (-b+a x^6\right )^{3/4}} \\ & = -\frac {2 b \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{4},\frac {5}{6},\frac {a x^6}{b}\right )}{x \left (-b+a x^6\right )^{3/4}}+\frac {2 c^2 x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+\frac {a x^5 \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{6},\frac {11}{6},\frac {a x^6}{b}\right )}{5 \left (-b+a x^6\right )^{3/4}}-(6 b c) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx-\frac {\left (2 b c^2\right ) \int \frac {1}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (2 c^3\right ) \int \frac {x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a}-\frac {\left (4 c \sqrt {\frac {a x^6}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a x^3} \\ & = -\frac {2 c \sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}-\frac {2 b \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{4},\frac {5}{6},\frac {a x^6}{b}\right )}{x \left (-b+a x^6\right )^{3/4}}+\frac {2 c^2 x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (-b+a x^6\right )^{3/4}}+\frac {a x^5 \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{6},\frac {11}{6},\frac {a x^6}{b}\right )}{5 \left (-b+a x^6\right )^{3/4}}-(6 b c) \int \frac {x^2}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx-\frac {\left (2 b c^2\right ) \int \frac {1}{\left (b-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, dx}{a}-\frac {\left (2 c^3\right ) \int \frac {x^4}{\left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-c x^4+a x^6\right )}{x^2 \left (-b+a x^6\right )^{3/4} \left (-b+c x^4+a x^6\right )} \, dx=\frac {2 \sqrt [4]{-b+a x^6}}{x}+\sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}{-\sqrt {c} x^2+\sqrt {-b+a x^6}}\right )-\sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^6}}\right ) \]

[In]

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

[Out]

(2*(-b + a*x^6)^(1/4))/x + Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqr
t[-b + a*x^6])] - Sqrt[2]*c^(1/4)*ArcTanh[(Sqrt[c]*x^2 + Sqrt[-b + a*x^6])/(Sqrt[2]*c^(1/4)*x*(-b + a*x^6)^(1/
4))]

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(\frac {-\ln \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}-b}}{\sqrt {a \,x^{6}-b}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right ) c^{\frac {1}{4}} \sqrt {2}\, x -2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {1}{4}} \sqrt {2}\, x -2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}-b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {1}{4}} \sqrt {2}\, x +4 \left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2 x}\)	\(187\)

[In]

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

[Out]

1/2*(-ln(((a*x^6-b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^6-b)^(1/2))/((a*x^6-b)^(1/2)-(a*x^6-b)^(1/4)*x*c^
(1/4)*2^(1/2)+c^(1/2)*x^2))*c^(1/4)*2^(1/2)*x-2*arctan((2^(1/2)*(a*x^6-b)^(1/4)+c^(1/4)*x)/c^(1/4)/x)*c^(1/4)*
2^(1/2)*x-2*arctan((2^(1/2)*(a*x^6-b)^(1/4)-c^(1/4)*x)/c^(1/4)/x)*c^(1/4)*2^(1/2)*x+4*(a*x^6-b)^(1/4))/x

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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