Optimal. Leaf size=216 \[ \frac {39 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 a^{17/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {78 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^2}+\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a} \]
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Rubi [A] time = 0.32, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 2024, 2011, 329, 220} \[ \frac {234 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^3}+\frac {39 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 a^{17/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {78 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^4}-\frac {26 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^2}+\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2024
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}-\frac {(13 b) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 a}\\ &=-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {\left (117 b^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{55 a^2}\\ &=\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}-\frac {\left (117 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^3}\\ &=-\frac {78 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {\left (39 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^4}\\ &=-\frac {78 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {\left (39 b^4 \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 )}{77 a^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {78 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {\left (78 b^4 \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 )}{77 a^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {78 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {39 b^{15/4} \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 )}{77 a^{17/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 124, normalized size = 0.57 \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}} \left (77 a^4 x^{8/3}-14 a^3 b x^2+26 a^2 b^2 x^{4/3}+195 b^4 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )-78 a b^3 x^{2/3}-195 b^4\right )}{385 a^4 \left (a x^{2/3}+b\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 3.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{3} - a b x^{\frac {7}{3}} + b^{2} x^{\frac {5}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{2} + b^{3}}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 163, normalized size = 0.75 \[ \frac {154 a^{5} x^{3}-28 a^{4} b \,x^{\frac {7}{3}}+52 a^{3} b^{2} x^{\frac {5}{3}}-156 a^{2} b^{3} x -390 a \,b^{4} x^{\frac {1}{3}}+195 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b^{4} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{385 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5}} \]
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 \[ \int \frac {x^{2}}{\sqrt {a x + b x^{\frac {1}{3}}}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {a\,x+b\,x^{1/3}}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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