Optimal. Leaf size=255 \[ \frac {b^5 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{25 \left (a+b x^3\right )}+\frac {5 a b^4 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {a^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \]
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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} \[ \frac {b^5 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{25 \left (a+b x^3\right )}+\frac {5 a b^4 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {a^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^9 \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^9 \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^9+5 a^4 b^6 x^{12}+10 a^3 b^7 x^{15}+10 a^2 b^8 x^{18}+5 a b^9 x^{21}+b^{10} x^{24}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {a^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {5 a b^4 x^{22} \sqrt {a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac {b^5 x^{25} \sqrt {a^2+2 a b x^3+b^2 x^6}}{25 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ \frac {x^{10} \sqrt {\left (a+b x^3\right )^2} \left (54340 a^5+209000 a^4 b x^3+339625 a^3 b^2 x^6+286000 a^2 b^3 x^9+123500 a b^4 x^{12}+21736 b^5 x^{15}\right )}{543400 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 57, normalized size = 0.22 \[ \frac {1}{25} \, b^{5} x^{25} + \frac {5}{22} \, a b^{4} x^{22} + \frac {10}{19} \, a^{2} b^{3} x^{19} + \frac {5}{8} \, a^{3} b^{2} x^{16} + \frac {5}{13} \, a^{4} b x^{13} + \frac {1}{10} \, a^{5} x^{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 105, normalized size = 0.41 \[ \frac {1}{25} \, b^{5} x^{25} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{22} \, a b^{4} x^{22} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{19} \, a^{2} b^{3} x^{19} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{8} \, a^{3} b^{2} x^{16} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{13} \, a^{4} b x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{10} \, a^{5} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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 \[ \frac {\left (21736 b^{5} x^{15}+123500 a \,b^{4} x^{12}+286000 a^{2} b^{3} x^{9}+339625 a^{3} b^{2} x^{6}+209000 a^{4} b \,x^{3}+54340 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x^{10}}{543400 \left (b \,x^{3}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 57, normalized size = 0.22 \[ \frac {1}{25} \, b^{5} x^{25} + \frac {5}{22} \, a b^{4} x^{22} + \frac {10}{19} \, a^{2} b^{3} x^{19} + \frac {5}{8} \, a^{3} b^{2} x^{16} + \frac {5}{13} \, a^{4} b x^{13} + \frac {1}{10} \, a^{5} x^{10} \]
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 \[ \int x^9\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \]
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 \[ \int x^{9} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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