Optimal. Leaf size=204 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}-\frac {1}{4} \text {RootSum}\left [a^6-2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(870\) vs. \(2(204)=408\).
time = 0.89, antiderivative size = 870, normalized size of antiderivative = 4.26, number of steps
used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2081, 6857, 61,
926, 93} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log (x)}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {2} \sqrt {a} x+\sqrt {b}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{a \sqrt [3]{x}}-1\right )}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 61
Rule 93
Rule 926
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-b+a x^2}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b}{2 x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 227, normalized size = 1.11 \begin {gather*} \frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )-a \text {RootSum}\left [a^6-2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 a \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (2 a \,x^{2}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.46, size = 2150, normalized size = 10.54 \begin {gather*} \text {Too large to display} \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 {a x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{2} - b\right )}\, dx \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*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {b-a\,x^2}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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