Optimal. Leaf size=61 \[ \frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}} \]
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Rubi [A] time = 0.09, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \[ \frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{b x^{2/3}}-\frac {a \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{2 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{b x^{2/3}}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{b x^{2/3}}+\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 90, normalized size = 1.48 \[ \frac {6 a \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right ) \left (\frac {\tanh ^{-1}\left (\sqrt {\frac {a \sqrt [3]{x}}{b}+1}\right )}{2 \sqrt {\frac {a \sqrt [3]{x}}{b}+1}}-\frac {b}{2 a \sqrt [3]{x}}\right )}{b^2 \sqrt {x^{2/3} \left (a \sqrt [3]{x}+b\right )}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 51, normalized size = 0.84 \[ -\frac {3 \, {\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {\sqrt {a x^{\frac {1}{3}} + b} a}{b x^{\frac {1}{3}}}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 1.00 \[ \frac {3 \sqrt {a \,x^{\frac {1}{3}}+b}\, \left (a b \,x^{\frac {1}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-\sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {3}{2}}\right )}{\sqrt {a x +b \,x^{\frac {2}{3}}}\, b^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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