Optimal. Leaf size=111 \[ \frac {\text {RootSum}\left [2 \text {$\#$1}^{18}-12 \text {$\#$1}^{15} a^3+30 \text {$\#$1}^{12} a^6-40 \text {$\#$1}^9 a^9+30 \text {$\#$1}^6 a^{12}-12 \text {$\#$1}^3 a^{15}+2 a^{18}-a^6 b^{11}\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{12 b} \]
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Rubi [B] time = 4.33, antiderivative size = 2286, normalized size of antiderivative = 20.59, number of steps used = 13, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 6725, 91}
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Warning: Unable to verify antiderivative.
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Rule 91
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, 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 (-2 b+a^6 x^6\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 {2} \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}-a^3 x^3\right )}-\frac {1}{2 \sqrt {2} \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}+a^3 x^3\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} \left (\sqrt {2} \sqrt {b}-a^3 x^3\right )} \, dx}{2 \sqrt {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} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}+a^3 x^3\right )} \, dx}{2 \sqrt {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 \left (-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {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 \left (\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {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 [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \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 [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \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 [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \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 [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \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 [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \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 [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{2} \sqrt [6]{b}-a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{2} \sqrt [6]{b}-a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [18]{2} \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}}}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [B] time = 0.49, size = 301, normalized size = 2.71 \begin {gather*} -\frac {x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2-2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2-\sqrt [3]{-1} 2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+\sqrt [3]{-1} 2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2-(-1)^{2/3} 2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (2 a^2+(-1)^{2/3} 2^{5/6} b^{11/6}\right ) x}{2 \left (x a^3+b^2\right )}\right )\right )}{4 b \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 15.56, size = 111, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [2 a^{18}-a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b} \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 {1}{{\left (a^{6} x^{6} - 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}} \left (a^{6} x^{6}-2 b \right )}\, dx\]
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 {1}{{\left (a^{6} x^{6} - 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}\,{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 {1}{\left (2\,b-a^6\,x^6\right )\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}} \,d x \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^{6} x^{6} - 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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