Optimal. Leaf size=193 \[ \frac {\sqrt {2} \left (8 a+\sqrt {8 a+1}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-4 a-\sqrt {8 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {8 a+1} \sqrt {-4 a-\sqrt {8 a+1}-1}}+\frac {\sqrt {2} \left (-8 a+\sqrt {8 a+1}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-4 a+\sqrt {8 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {8 a+1} \sqrt {-4 a+\sqrt {8 a+1}-1}} \]
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Rubi [A] time = 0.47, antiderivative size = 73, normalized size of antiderivative = 0.38, number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {6715, 826, 1161, 618, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\sqrt {8 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 {8 a+1}\right )}{5 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 826
Rule 1161
Rule 6715
Rubi steps
\begin {align*} \int \frac {x^4 \left (3+x^5\right )}{\sqrt {1+x^5} \left (-1+a-(1+2 a) x^5+a x^{10}\right )} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {3+x}{\sqrt {1+x} \left (-1+a+(-1-2 a) x+a x^2\right )} \, dx,x,x^5\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {2+x^2}{4 a+(-1-4 a) x^2+a x^4} \, dx,x,\sqrt {1+x^5}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{2-\frac {\sqrt {1+8 a} x}{\sqrt {a}}+x^2} \, dx,x,\sqrt {1+x^5}\right )}{5 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{2+\frac {\sqrt {1+8 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+8 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+8 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )}{5 a}\\ &=\frac {2 \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+8 a}}{\sqrt {a}}-2 \sqrt {1+x^5}\right )\right )}{5 \sqrt {a}}-\frac {2 \tanh ^{-1}\left (\sqrt {a} \left (\frac {\sqrt {1+8 a}}{\sqrt {a}}+2 \sqrt {1+x^5}\right )\right )}{5 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 158, normalized size = 0.82 \begin {gather*} \frac {\sqrt {4 a-\sqrt {8 a+1}+1} \left (\sqrt {8 a+1}+1\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {4 a-\sqrt {8 a+1}+1}}\right )-\left (\sqrt {8 a+1}-1\right ) \sqrt {4 a+\sqrt {8 a+1}+1} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {4 a+\sqrt {8 a+1}+1}}\right )}{10 \sqrt {2} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 193, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \left (8 a+\sqrt {8 a+1}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-4 a-\sqrt {8 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {8 a+1} \sqrt {-4 a-\sqrt {8 a+1}-1}}+\frac {\sqrt {2} \left (-8 a+\sqrt {8 a+1}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5+1}}{\sqrt {-4 a+\sqrt {8 a+1}-1}}\right )}{5 \sqrt {a} \sqrt {8 a+1} \sqrt {-4 a+\sqrt {8 a+1}-1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 92, normalized size = 0.48 \begin {gather*} \left [\frac {\log \left (\frac {a x^{10} - {\left (2 \, a - 1\right )} x^{5} - 2 \, \sqrt {x^{5} + 1} {\left (x^{5} - 1\right )} \sqrt {a} + a + 1}{a x^{10} - {\left (2 \, a + 1\right )} x^{5} + a - 1}\right )}{5 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {{\left (x^{5} - 1\right )} \sqrt {-a}}{\sqrt {x^{5} + 1}}\right )}{5 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (x^{5}+3\right )}{\sqrt {x^{5}+1}\, \left (-1+a -\left (1+2 a \right ) x^{5}+a \,x^{10}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} + 3\right )} x^{4}}{{\left (a x^{10} - {\left (2 \, a + 1\right )} x^{5} + a - 1\right )} \sqrt {x^{5} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 73, normalized size = 0.38 \begin {gather*} \frac {\ln \left (\frac {a-2\,a\,x^5+a\,x^{10}+2\,\sqrt {a}\,\sqrt {x^5+1}+x^5-2\,\sqrt {a}\,x^5\,\sqrt {x^5+1}+1}{2\,a\,x^5-a-a\,x^{10}+x^5+1}\right )}{5\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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