Optimal. Leaf size=126 \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}}}{a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{5/4} \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.12, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2010, 2013, 2011, 329, 220} \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}}}{a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{a^{5/4} \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2010
Rule 2011
Rule 2013
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \sqrt [3]{x}+a x}} \, dx &=\frac {2 \sqrt {b \sqrt [3]{x}+a x}}{a}-\frac {b \int \frac {1}{x^{2/3} \sqrt {b \sqrt [3]{x}+a x}} \, dx}{3 a}\\ &=\frac {2 \sqrt {b \sqrt [3]{x}+a x}}{a}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{a}\\ &=\frac {2 \sqrt {b \sqrt [3]{x}+a x}}{a}-\frac {\left (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 )}{a \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {2 \sqrt {b \sqrt [3]{x}+a x}}{a}-\frac {\left (2 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 )}{a \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {2 \sqrt {b \sqrt [3]{x}+a x}}{a}-\frac {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 )}{a^{5/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 80, normalized size = 0.63 \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}} \left (-b \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )+a x^{2/3}+b\right )}{a \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} - a b x^{\frac {4}{3}} + b^{2} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{3} + b^{3} x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 127, normalized size = 1.01 \[ \frac {2 a^{2} x +2 a b \,x^{\frac {1}{3}}-\sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2}} \]
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 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 40, normalized size = 0.32 \[ \frac {2\,x\,\sqrt {\frac {b}{a\,x^{2/3}}+1}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ -\frac {b}{a\,x^{2/3}}\right )}{\sqrt {a\,x+b\,x^{1/3}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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