Optimal. Leaf size=323 \[ -\frac {2 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {4 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{3/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {4 b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{15 a}+\frac {2}{3} x \sqrt {a x+b \sqrt [3]{x}} \]
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Rubi [A] time = 0.33, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2004, 2018, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {4 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{3/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4 b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{15 a}+\frac {2}{3} x \sqrt {a x+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2004
Rule 2018
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt {b \sqrt [3]{x}+a x} \, dx &=\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}+\frac {1}{9} (2 b) \int \frac {\sqrt [3]{x}}{\sqrt {b \sqrt [3]{x}+a x}} \, dx\\ &=\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}+\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a}+\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 a}\\ &=\frac {4 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a}+\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (2 b^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 a \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {4 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a}+\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (4 b^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 a \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {4 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a}+\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (4 b^{5/2} \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 )}{5 a^{3/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (4 b^{5/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 a^{3/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 a^{3/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {4 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a}+\frac {2}{3} x \sqrt {b \sqrt [3]{x}+a x}+\frac {4 b^{9/4} \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} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {2 b^{9/4} \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 )}{5 a^{7/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 94, normalized size = 0.29 \[ \frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}} \left (\left (a x^{2/3}+b\right ) \sqrt {\frac {a x^{2/3}}{b}+1}-b \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{3 a \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 8.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a x + b x^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \sqrt {a x + b x^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 207, normalized size = 0.64 \[ \frac {2 \sqrt {a x +b \,x^{\frac {1}{3}}}\, x}{3}+\frac {4 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b \,x^{\frac {1}{3}}}{15 a}-\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right ) b^{2}}{5 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{2}} \]
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 \[ \int \sqrt {a x + b x^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 40, normalized size = 0.12 \[ \frac {6\,x\,\sqrt {a\,x+b\,x^{1/3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ -\frac {a\,x^{2/3}}{b}\right )}{7\,\sqrt {\frac {a\,x^{2/3}}{b}+1}} \]
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 \[ \int \sqrt {a x + b \sqrt [3]{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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