Optimal. Leaf size=158 \[ -\frac {4 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a x+b x^3}}+\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {12}{77} a x^2 \sqrt {a x+b x^3} \]
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Rubi [A] time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2004, 2021, 2024, 2011, 329, 220} \[ -\frac {4 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a x+b x^3}}+\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2004
Rule 2011
Rule 2021
Rule 2024
Rubi steps
\begin {align*} \int \left (a x+b x^3\right )^{3/2} \, dx &=\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {1}{11} (6 a) \int x \sqrt {a x+b x^3} \, dx\\ &=\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}+\frac {1}{77} \left (12 a^2\right ) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{77 b}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{77 b \sqrt {a x+b x^3}}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {\left (8 a^3 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{77 b \sqrt {a x+b x^3}}\\ &=\frac {8 a^2 \sqrt {a x+b x^3}}{77 b}+\frac {12}{77} a x^2 \sqrt {a x+b x^3}+\frac {2}{11} x \left (a x+b x^3\right )^{3/2}-\frac {4 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 83, normalized size = 0.53 \[ \frac {2 \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right )^2 \sqrt {\frac {b x^2}{a}+1}-a^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{11 b \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{3} + a x\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (b x^{3} + a x\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 166, normalized size = 1.05 \[ \frac {2 \sqrt {b \,x^{3}+a x}\, b \,x^{4}}{11}+\frac {26 \sqrt {b \,x^{3}+a x}\, a \,x^{2}}{77}-\frac {4 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{3} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 \sqrt {b \,x^{3}+a x}\, b^{2}}+\frac {8 \sqrt {b \,x^{3}+a x}\, a^{2}}{77 b} \]
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 \[ \int {\left (b x^{3} + a x\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.00, size = 40, normalized size = 0.25 \[ \frac {2\,x\,{\left (b\,x^3+a\,x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {5}{4};\ \frac {9}{4};\ -\frac {b\,x^2}{a}\right )}{5\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
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 \[ \int \left (a x + b x^{3}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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