Optimal. Leaf size=107 \[ \frac {\text {RootSum}\left [\text {$\#$1}^{18}-6 \text {$\#$1}^{15} a^3+15 \text {$\#$1}^{12} a^6-20 \text {$\#$1}^9 a^9+15 \text {$\#$1}^6 a^{12}-6 \text {$\#$1}^3 a^{15}+a^{18}-2 a^6 b^{11}\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 b} \]
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Rubi [B] time = 3.23, antiderivative size = 2108, normalized size of antiderivative = 19.70, number of steps used = 13, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, 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 (-b+2 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 (-b+2 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 {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {b}-\sqrt {2} a^3 x^3\right )}-\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {b}+\sqrt {2} 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 {b}-\sqrt {2} a^3 x^3\right )} \, 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} \sqrt [3]{b^2+a^3 x} \left (\sqrt {b}+\sqrt {2} a^3 x^3\right )} \, 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 \left (-\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, 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 \left (\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, 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 [6]{b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{b}-\sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{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]{b}-\sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{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]{b}-\sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{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]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{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]{b}+\sqrt [3]{-1} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{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]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{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]{b}-(-1)^{2/3} \sqrt [6]{2} a x\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} \sqrt [6]{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 [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}}}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 271, normalized size = 2.53 \begin {gather*} -\frac {x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-\sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+\sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+\sqrt [3]{-1} \sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-(-1)^{2/3} \sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+(-1)^{2/3} \sqrt [6]{2} b^{11/6}\right ) x}{x a^3+b^2}\right )\right )}{2 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 = 0.00, size = 107, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a^{18}-2 a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 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 (2 \, a^{6} x^{6} - 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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}} \left (2 a^{6} x^{6}-b \right )}\, dx\]
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 (2 \, a^{6} x^{6} - 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 (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a^6\,x^6\right )} \,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 (2 a^{6} x^{6} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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