Optimal. Leaf size=52 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5-x}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5-x}}\right )}{\sqrt [4]{a}} \]
________________________________________________________________________________________
Rubi [F] time = 1.51, 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 {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx &=\int \frac {x \left (1+3 x^4\right )}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {\sqrt {x} \left (1+3 x^4\right )}{\sqrt {-1+x^4} \left (a-x^2-2 a x^4+a x^8\right )} \, dx}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+3 x^8\right )}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a x^{16}\right )}+\frac {3 x^{10}}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.67, size = 52, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5-x}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5-x}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.52, size = 202, normalized size = 3.88 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x^{5} - x}}{{\left (x^{4} - 1\right )} a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} - \frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} + 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} - 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + x}{{\left (a x^{8} - 2 \, a x^{4} - x^{2} + a\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{5}+x}{\sqrt {x^{5}-x}\, \left (a \,x^{8}-2 a \,x^{4}-x^{2}+a \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + x}{{\left (a x^{8} - 2 \, a x^{4} - x^{2} + a\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.56, size = 103, normalized size = 1.98 \begin {gather*} \frac {\ln \left (\frac {x-2\,a^{1/4}\,\sqrt {x^5-x}-\sqrt {a}+\sqrt {a}\,x^4}{x+\sqrt {a}-\sqrt {a}\,x^4}\right )}{2\,a^{1/4}}+\frac {\ln \left (\frac {x+\sqrt {a}-\sqrt {a}\,x^4+a^{1/4}\,\sqrt {x^5-x}\,2{}\mathrm {i}}{x-\sqrt {a}+\sqrt {a}\,x^4}\right )\,1{}\mathrm {i}}{2\,a^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________