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

Optimal. Leaf size=95 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {2 \left (a x^6-b\right )^{3/4}}{3 x^3} \]

________________________________________________________________________________________

Rubi [F]  time = 3.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-b+a x^6}}+\frac {2 b}{x^4 \sqrt [4]{-b+a x^6}}+\frac {a x^2}{\sqrt [4]{-b+a x^6}}+\frac {-3 b-2 x^4}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}\right ) \, dx\\ &=a \int \frac {x^2}{\sqrt [4]{-b+a x^6}} \, dx+(2 b) \int \frac {1}{x^4 \sqrt [4]{-b+a x^6}} \, dx+\int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx+\int \frac {-3 b-2 x^4}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx\\ &=\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )+\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )+\frac {\sqrt [4]{1-\frac {a x^6}{b}} \int \frac {1}{\sqrt [4]{1-\frac {a x^6}{b}}} \, dx}{\sqrt [4]{-b+a x^6}}+\int \left (-\frac {3 b}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}+\frac {2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx\\ &=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^2}} \, dx,x,x^3\right )-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx+\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}\\ &=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx-\frac {\left (2 \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}-\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {b}}}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}\\ &=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {2 a x^3 \sqrt [4]{-b+a x^6}}{3 \left (\sqrt {b}+\sqrt {-b+a x^6}\right )}-\frac {2 \sqrt [4]{b} \sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 x^3}+\frac {\sqrt [4]{b} \sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 x^3}+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx-\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}+\frac {\left (2 \sqrt {b} \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {b}}}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 x^3}\\ &=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 15.44, size = 95, normalized size = 1.00 \begin {gather*} \frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-b + a*x^6)^(3/4))/(3*x^3) + ArcTan[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4) + ArcTanh[(2^(1/4)
*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{6}+2 b \right ) \left (a \,x^{6}-x^{4}-b \right )}{x^{4} \left (a \,x^{6}-b \right )^{\frac {1}{4}} \left (a \,x^{6}-2 x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,x^6+2\,b\right )\,\left (-a\,x^6+x^4+b\right )}{x^4\,{\left (a\,x^6-b\right )}^{1/4}\,\left (-a\,x^6+2\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________