Optimal. Leaf size=137 \[ -\frac {2 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a x+b x^3}}+\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3} \]
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Rubi [A] time = 0.12, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2021, 2024, 2011, 329, 220} \[ -\frac {2 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {a x+b x^3}}+\frac {2}{7} x^2 \sqrt {a x+b x^3}+\frac {4 a \sqrt {a x+b x^3}}{21 b} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2021
Rule 2024
Rubi steps
\begin {align*} \int x \sqrt {a x+b x^3} \, dx &=\frac {2}{7} x^2 \sqrt {a x+b x^3}+\frac {1}{7} (2 a) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{21 b}\\ &=\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (2 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{21 b \sqrt {a x+b x^3}}\\ &=\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {\left (4 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{21 b \sqrt {a x+b x^3}}\\ &=\frac {4 a \sqrt {a x+b x^3}}{21 b}+\frac {2}{7} x^2 \sqrt {a x+b x^3}-\frac {2 a^{7/4} \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 )}{21 b^{5/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 79, normalized size = 0.58 \[ \frac {2 \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1}-a \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{7 b \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{3} + a x} x, x\right ) \]
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 \[ \int \sqrt {b x^{3} + a x} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 146, normalized size = 1.07 \[ \frac {2 \sqrt {b \,x^{3}+a x}\, x^{2}}{7}-\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{2} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 \sqrt {b \,x^{3}+a x}\, b^{2}}+\frac {4 \sqrt {b \,x^{3}+a x}\, a}{21 b} \]
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 \[ \int \sqrt {b x^{3} + a x} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {b\,x^3+a\,x} \,d x \]
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 \[ \int x \sqrt {x \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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