Optimal. Leaf size=226 \[ -\frac {1}{3} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\& ,\frac {-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-a b^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 a^3 \log (x) \text {$\#$1}^3-2 a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^6+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a^6 \text {$\#$1}-2 a^3 \text {$\#$1}^4+\text {$\#$1}^7}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1265\) vs. \(2(226)=452\).
time = 1.18, antiderivative size = 1265, normalized size of antiderivative = 5.60, number of steps
used = 6, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2081, 6857, 93}
\begin {gather*} -\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 93
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (-b+2 a x^3\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^3\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 {-\frac {\sqrt [3]{-1} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\frac {\sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\frac {(-1)^{2/3} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A]
time = 10.08, size = 230, normalized size = 1.02 \begin {gather*} \frac {x \left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x+\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (2 a \,x^{3}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 = 108.94, size = 149068, normalized size = 659.59 \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^{3} - 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^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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