Optimal. Leaf size=255 \[ \frac {b^5 x^{28} \sqrt {a^2+2 a b x^3+b^2 x^6}}{28 \left (a+b x^3\right )}+\frac {a b^4 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a^5 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {5 a^4 b x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 270} \begin {gather*} \frac {b^5 x^{28} \sqrt {a^2+2 a b x^3+b^2 x^6}}{28 \left (a+b x^3\right )}+\frac {a b^4 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {5 a^4 b x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {a^5 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^{12} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^{12} \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x^{12}+5 a^4 b^6 x^{15}+10 a^3 b^7 x^{18}+10 a^2 b^8 x^{21}+5 a b^9 x^{24}+b^{10} x^{27}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {a^5 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {5 a^4 b x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a b^4 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b^5 x^{28} \sqrt {a^2+2 a b x^3+b^2 x^6}}{28 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} \frac {x^{13} \sqrt {\left (a+b x^3\right )^2} \left (117040 a^5+475475 a^4 b x^3+800800 a^3 b^2 x^6+691600 a^2 b^3 x^9+304304 a b^4 x^{12}+54340 b^5 x^{15}\right )}{1521520 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 24.78, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (117040 a^5 x^{13}+475475 a^4 b x^{16}+800800 a^3 b^2 x^{19}+691600 a^2 b^3 x^{22}+304304 a b^4 x^{25}+54340 b^5 x^{28}\right )}{1521520 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.99, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{28} \, b^{5} x^{28} + \frac {1}{5} \, a b^{4} x^{25} + \frac {5}{11} \, a^{2} b^{3} x^{22} + \frac {10}{19} \, a^{3} b^{2} x^{19} + \frac {5}{16} \, a^{4} b x^{16} + \frac {1}{13} \, a^{5} x^{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 105, normalized size = 0.41 \begin {gather*} \frac {1}{28} \, b^{5} x^{28} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{5} \, a b^{4} x^{25} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{11} \, a^{2} b^{3} x^{22} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{19} \, a^{3} b^{2} x^{19} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{16} \, a^{4} b x^{16} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{13} \, a^{5} x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.31 \begin {gather*} \frac {\left (54340 b^{5} x^{15}+304304 a \,b^{4} x^{12}+691600 a^{2} b^{3} x^{9}+800800 a^{3} b^{2} x^{6}+475475 a^{4} b \,x^{3}+117040 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x^{13}}{1521520 \left (b \,x^{3}+a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{28} \, b^{5} x^{28} + \frac {1}{5} \, a b^{4} x^{25} + \frac {5}{11} \, a^{2} b^{3} x^{22} + \frac {10}{19} \, a^{3} b^{2} x^{19} + \frac {5}{16} \, a^{4} b x^{16} + \frac {1}{13} \, a^{5} x^{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{12}\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \end {gather*}
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 \begin {gather*} \int x^{12} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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