Optimal. Leaf size=94 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3+b}}{\sqrt {a x^3+b}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3+b}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3+b}}\right ) \]
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Rubi [F] time = 0.72, 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 {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx &=\int \left (\frac {4 b}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )}+\frac {a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )}\right ) \, dx\\ &=a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.77, size = 94, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{b+a x^3}}{x^2+\sqrt {b+a x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} + 4 \, b}{{\left (a x^{3} + x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+4 b}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (a \,x^{3}+x^{4}+b \right )}\, dx\]
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 {a x^{3} + 4 \, b}{{\left (a x^{3} + x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^3+4\,b}{{\left (a\,x^3+b\right )}^{1/4}\,\left (x^4+a\,x^3+b\right )} \,d x \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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