Optimal. Leaf size=69 \[ \frac {\text {RootSum}\left [a^6-a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{2 b} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(624\) vs. \(2(69)=138\).
time = 0.17, antiderivative size = 624, normalized size of antiderivative = 9.04, number of steps
used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2081, 926, 93}
\begin {gather*} \frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}-b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}+b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {b}-\sqrt {a} x\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {a} x+\sqrt {b}\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}}}-\sqrt [3]{x}\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}}}-\sqrt [3]{x}\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 926
Rule 2081
Rubi steps
\begin {align*} \int \frac {1}{\left (-b+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 {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+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 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}\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} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \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} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \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}-b^{3/2}} \sqrt [3]{x}}\right )}{2 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-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}+b^{3/2}} \sqrt [3]{x}}\right )}{2 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+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 {a} x\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+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 {a} x\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-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}-b^{3/2}}}\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}-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}+b^{3/2}}}\right )}{4 \sqrt [6]{a} b \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 106, normalized size = 1.54 \begin {gather*} \frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^6-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 ]}{2 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{2}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 = 0.43, size = 2012, normalized size = 29.16 \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 {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (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 = -0.01 \begin {gather*} -\int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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