Optimal. Leaf size=123 \[ \frac {2 b^{3/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x+b \sqrt [3]{x}}}+2 \sqrt {a x+b \sqrt [3]{x}} \]
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Rubi [A] time = 0.15, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 2021, 2011, 329, 220} \[ \frac {2 b^{3/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x+b \sqrt [3]{x}}}+2 \sqrt {a x+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2021
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x} \, dx &=3 \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt {b \sqrt [3]{x}+a x}+(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (2 b \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {b \sqrt [3]{x}+a x}}\\ &=2 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (4 b \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt {b \sqrt [3]{x}+a x}}\\ &=2 \sqrt {b \sqrt [3]{x}+a x}+\frac {2 b^{3/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 54, normalized size = 0.44 \[ \frac {6 \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )}{\sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x}, x\right ) \]
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 \[ \int \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 1.07 \[ \frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a x +b \,x^{\frac {1}{3}}}\, a}+2 \sqrt {a x +b \,x^{\frac {1}{3}}} \]
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 \[ \int \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a\,x+b\,x^{1/3}}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a x + b \sqrt [3]{x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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