Optimal. Leaf size=215 \[ \frac {\left (a^3 x^3+b^2 x^2\right )^{2/3} \left (3 a^3 x-4 b^2\right )}{6 a^5 x}+\frac {\left (-9 a^5 b-2 b^4\right ) \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{9 a^6}+\frac {\left (9 a^5 b+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{3 \sqrt {3} a^6}+\frac {\left (9 a^5 b+2 b^4\right ) \log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{18 a^6} \]
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Rubi [B] time = 0.36, antiderivative size = 467, normalized size of antiderivative = 2.17, number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2053, 2011, 59, 2024} \begin {gather*} -\frac {b x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 b x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} b x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{9 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{3 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^2 \left (a^3 x^3+b^2 x^2\right )^{2/3}}{3 a^5 x}+\frac {\left (a^3 x^3+b^2 x^2\right )^{2/3}}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2011
Rule 2024
Rule 2053
Rubi steps
\begin {align*} \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\int \left (\frac {b}{\sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}}\right ) \, dx\\ &=a \int \frac {x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx+b \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {\left (2 b^2\right ) \int \frac {x}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{3 a^2}+\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}} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}-\frac {\sqrt {3} b 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 )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 b 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 )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2 b^4\right ) \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{9 a^5}\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}-\frac {\sqrt {3} b 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 )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 b 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 )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2 b^4 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}{9 a^5 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}-\frac {\sqrt {3} b 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 )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {2 b^4 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 )}{3 \sqrt {3} a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b^4 x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{9 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 b 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 )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b^4 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 )}{3 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 117, normalized size = 0.54 \begin {gather*} \frac {3 \left (x^2 \left (a^3 x+b^2\right )\right )^{2/3} \left (b^3 \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )-2 b^3 \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )+\left (a^5+b^3\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )\right )}{a^5 b x \left (\frac {a^3 x}{b^2}+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 221, normalized size = 1.03 \begin {gather*} \frac {\left (-4 b^2+3 a^3 x\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{6 a^5 x}+\frac {\left (9 a^5 b+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{3 \sqrt {3} a^6}+\frac {\left (-9 a^5 b-2 b^4\right ) \log \left (a^7 x-a^6 \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{9 a^6}+\frac {\left (9 a^5 b+2 b^4\right ) \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 )}{18 a^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 206, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + 2 \, {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (3 \, a^{4} x - 4 \, a b^{2}\right )}}{18 \, a^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 195, normalized size = 0.91 \begin {gather*} -\frac {\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{6}} - \frac {{\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{a^{6}} + \frac {2 \, {\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{6}} - \frac {3 \, {\left (7 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}} a^{3} b^{6} - 4 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {5}{3}} b^{6}\right )} x^{2}}{a^{5} b^{4}}}{18 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+b}{\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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*} \int \frac {a x^{2} + b}{{\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 = -0.00 \begin {gather*} \int \frac {a\,x^2+b}{{\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 {a x^{2} + b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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