Optimal. Leaf size=16 \[ -2 \tan ^{-1}\left (\frac {x}{\sqrt {a x^5+b}}\right ) \]
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Rubi [F] time = 0.84, 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 {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx &=\int \left (\frac {3}{\sqrt {b+a x^5}}-\frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx\\ &=3 \int \frac {1}{\sqrt {b+a x^5}} \, dx-\int \frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx\\ &=\frac {\left (3 \sqrt {1+\frac {a x^5}{b}}\right ) \int \frac {1}{\sqrt {1+\frac {a x^5}{b}}} \, dx}{\sqrt {b+a x^5}}-\int \left (\frac {5 b}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}+\frac {3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx\\ &=\frac {3 x \sqrt {1+\frac {a x^5}{b}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {6}{5};-\frac {a x^5}{b}\right )}{\sqrt {b+a x^5}}-3 \int \frac {x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx-(5 b) \int \frac {1}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.46, size = 16, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt {b+a x^5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 35, normalized size = 2.19 \begin {gather*} \arctan \left (\frac {{\left (a x^{5} - x^{2} + b\right )} \sqrt {a x^{5} + b}}{2 \, {\left (a x^{6} + b x\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, a x^{5} - 2 \, b}{{\left (a x^{5} + x^{2} + b\right )} \sqrt {a x^{5} + b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {3 a \,x^{5}-2 b}{\sqrt {a \,x^{5}+b}\, \left (a \,x^{5}+x^{2}+b \right )}\, dx\]
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 {3 \, a x^{5} - 2 \, b}{{\left (a x^{5} + x^{2} + b\right )} \sqrt {a x^{5} + b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 42, normalized size = 2.62 \begin {gather*} \ln \left (\frac {b+a\,x^5-x^2+x\,\sqrt {a\,x^5+b}\,2{}\mathrm {i}}{a\,x^5+x^2+b}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 a x^{5} - 2 b}{\sqrt {a x^{5} + b} \left (a x^{5} + b + x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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