Optimal. Leaf size=201 \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^9}{20 b^5}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{9 b^5}+\frac {3 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{8 b^5}+\frac {a^4 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^5}-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^5} \]
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Rubi [A] time = 0.13, antiderivative size = 201, 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 = {1111, 645} \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^9}{20 b^5}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{9 b^5}+\frac {3 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{8 b^5}-\frac {2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^5}+\frac {a^4 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^5} \]
Antiderivative was successfully verified.
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Rule 645
Rule 1111
Rubi steps
\begin {align*} \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (\frac {a^4 \left (a b+b^2 x\right )^5}{b^4}-\frac {4 a^3 \left (a b+b^2 x\right )^6}{b^5}+\frac {6 a^2 \left (a b+b^2 x\right )^7}{b^6}-\frac {4 a \left (a b+b^2 x\right )^8}{b^7}+\frac {\left (a b+b^2 x\right )^9}{b^8}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^4 \left (a+b x^2\right )^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 b^5}-\frac {2 a^3 \left (a+b x^2\right )^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 b^5}+\frac {3 a^2 \left (a+b x^2\right )^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 b^5}-\frac {2 a \left (a+b x^2\right )^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 b^5}+\frac {\left (a+b x^2\right )^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{20 b^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.41 \[ \frac {x^{10} \sqrt {\left (a+b x^2\right )^2} \left (252 a^5+1050 a^4 b x^2+1800 a^3 b^2 x^4+1575 a^2 b^3 x^6+700 a b^4 x^8+126 b^5 x^{10}\right )}{2520 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 57, normalized size = 0.28 \[ \frac {1}{20} \, b^{5} x^{20} + \frac {5}{18} \, a b^{4} x^{18} + \frac {5}{8} \, a^{2} b^{3} x^{16} + \frac {5}{7} \, a^{3} b^{2} x^{14} + \frac {5}{12} \, a^{4} b x^{12} + \frac {1}{10} \, a^{5} x^{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 105, normalized size = 0.52 \[ \frac {1}{20} \, b^{5} x^{20} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{18} \, a b^{4} x^{18} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{8} \, a^{2} b^{3} x^{16} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{7} \, a^{3} b^{2} x^{14} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{12} \, a^{4} b x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{10} \, a^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.40 \[ \frac {\left (126 b^{5} x^{10}+700 a \,b^{4} x^{8}+1575 a^{2} b^{3} x^{6}+1800 a^{3} b^{2} x^{4}+1050 a^{4} b \,x^{2}+252 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{10}}{2520 \left (b \,x^{2}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 57, normalized size = 0.28 \[ \frac {1}{20} \, b^{5} x^{20} + \frac {5}{18} \, a b^{4} x^{18} + \frac {5}{8} \, a^{2} b^{3} x^{16} + \frac {5}{7} \, a^{3} b^{2} x^{14} + \frac {5}{12} \, a^{4} b x^{12} + \frac {1}{10} \, a^{5} x^{10} \]
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 \[ \int x^9\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
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 \[ \int x^{9} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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