Optimal. Leaf size=193 \[ \frac {\sqrt {2} \left (4 a+\sqrt {4 a+1}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-2 a-\sqrt {4 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {4 a+1} \sqrt {-2 a-\sqrt {4 a+1}-1}}+\frac {\sqrt {2} \left (-4 a+\sqrt {4 a+1}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-2 a+\sqrt {4 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {4 a+1} \sqrt {-2 a+\sqrt {4 a+1}-1}} \]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 73, normalized size of antiderivative = 0.38, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6715, 826, 1161, 618, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\sqrt {4 a+1}-2 \sqrt {a} \sqrt {x^5+1}\right )}{5 \sqrt {a}}-\frac {2 \tanh ^{-1}\left (2 \sqrt {a} \sqrt {x^5+1}+\sqrt {4 a+1}\right )}{5 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 826
Rule 1161
Rule 6715
Rubi steps
\begin {align*} \int \frac {x^4 \left (2+x^5\right )}{\sqrt {1+x^5} \left (-1-x^5+a x^{10}\right )} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {2+x}{\sqrt {1+x} \left (-1-x+a x^2\right )} \, dx,x,x^5\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1+x^2}{a+(-1-2 a) x^2+a x^4} \, dx,x,\sqrt {1+x^5}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {\sqrt {1+4 a} x}{\sqrt {a}}+x^2} \, dx,x,\sqrt {1+x^5}\right )}{5 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {\sqrt {1+4 a} x}{\sqrt {a}}+x^2} \, dx,x,\sqrt {1+x^5}\right )}{5 a}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-x^2} \, dx,x,-\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )}{5 a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-x^2} \, dx,x,\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )}{5 a}\\ &=\frac {2 \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+4 a}}{\sqrt {a}}-2 \sqrt {1+x^5}\right )\right )}{5 \sqrt {a}}-\frac {2 \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+4 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )\right )}{5 \sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 158, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2 a-\sqrt {4 a+1}+1} \left (\sqrt {4 a+1}+1\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {2 a-\sqrt {4 a+1}+1}}\right )-\left (\sqrt {4 a+1}-1\right ) \sqrt {2 a+\sqrt {4 a+1}+1} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {2 a+\sqrt {4 a+1}+1}}\right )}{5 \sqrt {2} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.32, size = 193, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \left (1+4 a+\sqrt {1+4 a}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a-\sqrt {1+4 a}}}\right )}{5 \sqrt {a} \sqrt {1+4 a} \sqrt {-1-2 a-\sqrt {1+4 a}}}+\frac {\sqrt {2} \left (-1-4 a+\sqrt {1+4 a}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {1+x^5}}{\sqrt {-1-2 a+\sqrt {1+4 a}}}\right )}{5 \sqrt {a} \sqrt {1+4 a} \sqrt {-1-2 a+\sqrt {1+4 a}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 74, normalized size = 0.38 \begin {gather*} \left [\frac {\log \left (\frac {a x^{10} - 2 \, \sqrt {x^{5} + 1} \sqrt {a} x^{5} + x^{5} + 1}{a x^{10} - x^{5} - 1}\right )}{5 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{5}}{\sqrt {x^{5} + 1}}\right )}{5 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (x^{5}+2\right )}{\sqrt {x^{5}+1}\, \left (a \,x^{10}-x^{5}-1\right )}\, dx\]
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 {{\left (x^{5} + 2\right )} x^{4}}{{\left (a x^{10} - x^{5} - 1\right )} \sqrt {x^{5} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.29, size = 47, normalized size = 0.24 \begin {gather*} \frac {\ln \left (\frac {a\,x^{10}+x^5-2\,\sqrt {a}\,x^5\,\sqrt {x^5+1}+1}{-4\,a\,x^{10}+4\,x^5+4}\right )}{5\,\sqrt {a}} \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]
________________________________________________________________________________________