Optimal. Leaf size=255 \[ \frac {b^5 x^{21} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}+\frac {5 a b^4 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {a^5 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {2 a^3 b^2 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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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 = {1112, 270} \[ \frac {b^5 x^{21} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}+\frac {5 a b^4 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {2 a^3 b^2 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {a^5 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int x^{10} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^{10} \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 x^{10}+5 a^4 b^6 x^{12}+10 a^3 b^7 x^{14}+10 a^2 b^8 x^{16}+5 a b^9 x^{18}+b^{10} x^{20}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^5 x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {5 a^4 b x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {2 a^3 b^2 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {5 a b^4 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac {b^5 x^{21} \sqrt {a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ \frac {x^{11} \sqrt {\left (a+b x^2\right )^2} \left (88179 a^5+373065 a^4 b x^2+646646 a^3 b^2 x^4+570570 a^2 b^3 x^6+255255 a b^4 x^8+46189 b^5 x^{10}\right )}{969969 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 57, normalized size = 0.22 \[ \frac {1}{21} \, b^{5} x^{21} + \frac {5}{19} \, a b^{4} x^{19} + \frac {10}{17} \, a^{2} b^{3} x^{17} + \frac {2}{3} \, a^{3} b^{2} x^{15} + \frac {5}{13} \, a^{4} b x^{13} + \frac {1}{11} \, a^{5} x^{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 105, normalized size = 0.41 \[ \frac {1}{21} \, b^{5} x^{21} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{19} \, a b^{4} x^{19} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{17} \, a^{2} b^{3} x^{17} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, a^{3} b^{2} x^{15} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{13} \, a^{4} b x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{11} \, a^{5} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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 \[ \frac {\left (46189 b^{5} x^{10}+255255 a \,b^{4} x^{8}+570570 a^{2} b^{3} x^{6}+646646 a^{3} b^{2} x^{4}+373065 a^{4} b \,x^{2}+88179 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{11}}{969969 \left (b \,x^{2}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 57, normalized size = 0.22 \[ \frac {1}{21} \, b^{5} x^{21} + \frac {5}{19} \, a b^{4} x^{19} + \frac {10}{17} \, a^{2} b^{3} x^{17} + \frac {2}{3} \, a^{3} b^{2} x^{15} + \frac {5}{13} \, a^{4} b x^{13} + \frac {1}{11} \, a^{5} x^{11} \]
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 \[ \int x^{10}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
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 \[ \int x^{10} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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