Optimal. Leaf size=326 \[ \frac {7 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {14 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {14 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{5/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {14 b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{15 a^2}+\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{3 a} \]
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Rubi [A] time = 0.35, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2018, 2024, 2032, 329, 305, 220, 1196} \[ \frac {14 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{5/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {7 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {14 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {14 b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{15 a^2}+\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{3 a} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}-\frac {(7 b) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3 a}\\ &=-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}+\frac {\left (7 b^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 a^2}\\ &=-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}+\frac {\left (7 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^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}+\frac {\left (14 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^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}+\frac {\left (14 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^{5/2} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (14 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^{5/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {14 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 a^{5/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}-\frac {14 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^{11/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {7 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^{11/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 106, normalized size = 0.33 \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}} \left (5 a^2 x^{5/3}+7 b^2 \sqrt [3]{x} \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )-2 a b x-7 b^2 \sqrt [3]{x}\right )}{15 a^2 \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} - a b x^{\frac {4}{3}} + b^{2} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{2} + b^{3}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 228, normalized size = 0.70 \[ -\frac {-10 a^{3} x^{2}+4 a^{2} b \,x^{\frac {4}{3}}-42 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b^{3} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b^{3} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+14 a \,b^{2} x^{\frac {2}{3}}}{15 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3}} \]
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 \[ \int \frac {x}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{\sqrt {a\,x+b\,x^{1/3}}} \,d x \]
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 \[ \int \frac {x}{\sqrt {a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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