Optimal. Leaf size=186 \[ \frac {4 a^{15/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {4}{55} a x^4 \sqrt {a x+b x^3} \]
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Rubi [A] time = 0.23, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2021, 2024, 2011, 329, 220} \[ -\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {4 a^{15/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a x+b x^3}}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2021
Rule 2024
Rubi steps
\begin {align*} \int x^2 \left (a x+b x^3\right )^{3/2} \, dx &=\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {1}{5} (2 a) \int x^3 \sqrt {a x+b x^3} \, dx\\ &=\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {1}{55} \left (4 a^2\right ) \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx\\ &=\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{77 b}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (4 a^4\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{231 b^2}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (4 a^4 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {\left (8 a^4 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{231 b^2 \sqrt {a x+b x^3}}\\ &=-\frac {8 a^3 \sqrt {a x+b x^3}}{231 b^2}+\frac {8 a^2 x^2 \sqrt {a x+b x^3}}{385 b}+\frac {4}{55} a x^4 \sqrt {a x+b x^3}+\frac {2}{15} x^3 \left (a x+b x^3\right )^{3/2}+\frac {4 a^{15/4} \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 )}{231 b^{9/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 94, normalized size = 0.51 \[ \frac {2 \sqrt {x \left (a+b x^2\right )} \left (5 a^3 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )-\left (5 a-11 b x^2\right ) \left (a+b x^2\right )^2 \sqrt {\frac {b x^2}{a}+1}\right )}{165 b^2 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{5} + a x^{3}\right )} \sqrt {b x^{3} + a x}, x\right ) \]
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 \[ \int {\left (b x^{3} + a x\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 188, normalized size = 1.01 \[ \frac {2 \sqrt {b \,x^{3}+a x}\, b \,x^{6}}{15}+\frac {34 \sqrt {b \,x^{3}+a x}\, a \,x^{4}}{165}+\frac {8 \sqrt {b \,x^{3}+a x}\, a^{2} x^{2}}{385 b}+\frac {4 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a^{4} \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{231 \sqrt {b \,x^{3}+a x}\, b^{3}}-\frac {8 \sqrt {b \,x^{3}+a x}\, a^{3}}{231 b^{2}} \]
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 \[ \int {\left (b x^{3} + a x\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (b\,x^3+a\,x\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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