Optimal. Leaf size=208 \[ \frac {4 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 a^{9/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
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Rubi [A] time = 0.27, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2004, 2018, 2021, 2024, 2011, 329, 220} \[ \frac {4 b^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 a^{9/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2004
Rule 2011
Rule 2018
Rule 2021
Rule 2024
Rubi steps
\begin {align*} \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{5} (2 b) \int \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x} \, dx\\ &=\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{5} (6 b) \operatorname {Subst}\left (\int x^3 \sqrt {b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{55} \left (12 b^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a}\\ &=-\frac {8 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (4 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{77 a^2}\\ &=-\frac {8 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (4 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^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {8 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (8 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^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {8 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {4 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^{9/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 106, normalized size = 0.51 \[ \frac {2 \sqrt {a x+b \sqrt [3]{x}} \left (5 b^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )-\left (5 b-11 a x^{2/3}\right ) \left (a x^{2/3}+b\right )^2 \sqrt {\frac {a x^{2/3}}{b}+1}\right )}{55 a^2 \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 164, normalized size = 0.79 \[ \frac {\frac {2 a^{5} x^{3}}{5}+\frac {56 a^{4} b \,x^{\frac {7}{3}}}{55}+\frac {262 a^{3} b^{2} x^{\frac {5}{3}}}{385}-\frac {16 a^{2} b^{3} x}{385}-\frac {8 a \,b^{4} x^{\frac {1}{3}}}{77}+\frac {4 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, b^{4} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77}}{\sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3}} \]
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 \[ \int {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 40, normalized size = 0.19 \[ \frac {2\,x\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {9}{4};\ \frac {13}{4};\ -\frac {a\,x^{2/3}}{b}\right )}{3\,{\left (\frac {a\,x^{2/3}}{b}+1\right )}^{3/2}} \]
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 \[ \int \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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