Optimal. Leaf size=109 \[ -\frac {32 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{105 a^4 x}+\frac {16 b^2 \left (a x+b x^{2/3}\right )^{3/2}}{35 a^3 x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a} \]
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Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2002, 2016, 2014} \[ -\frac {32 b^3 \left (a x+b x^{2/3}\right )^{3/2}}{105 a^4 x}+\frac {16 b^2 \left (a x+b x^{2/3}\right )^{3/2}}{35 a^3 x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \sqrt {b x^{2/3}+a x} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac {(2 b) \int \frac {\sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{3 a}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac {4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x^{2/3}} \, dx}{21 a^2}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}+\frac {16 b^2 \left (b x^{2/3}+a x\right )^{3/2}}{35 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}-\frac {\left (16 b^3\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx}{105 a^3}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac {32 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{105 a^4 x}+\frac {16 b^2 \left (b x^{2/3}+a x\right )^{3/2}}{35 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.64 \[ \frac {2 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \left (35 a^3 x-30 a^2 b x^{2/3}+24 a b^2 \sqrt [3]{x}-16 b^3\right )}{105 a^4 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 143, normalized size = 1.31 \[ \frac {32 \, b^{\frac {9}{2}}}{105 \, a^{4}} + \frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} - 21 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{2} - 35 \, \sqrt {a x^{\frac {1}{3}} + b} b^{3}\right )} b}{a^{3}} + \frac {35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x^{\frac {1}{3}} + b} b^{4}}{a^{3}}\right )}}{105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.52 \[ -\frac {2 \sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (a \,x^{\frac {1}{3}}+b \right ) \left (-35 a^{3} x +30 a^{2} b \,x^{\frac {2}{3}}-24 a \,b^{2} x^{\frac {1}{3}}+16 b^{3}\right )}{105 a^{4} x^{\frac {1}{3}}} \]
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 \[ \int \sqrt {a x + b x^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 40, normalized size = 0.37 \[ \frac {3\,x\,\sqrt {a\,x+b\,x^{2/3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},4;\ 5;\ -\frac {a\,x^{1/3}}{b}\right )}{4\,\sqrt {\frac {a\,x^{1/3}}{b}+1}} \]
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 \[ \int \sqrt {a x + b x^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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