Optimal. Leaf size=255 \[ \frac {b^5 x^{22} \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {5 a^2 b^3 x^{18} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {a^5 x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 \left (a+b x^2\right )}+\frac {5 a^4 b x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \begin {gather*} \frac {b^5 x^{22} \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {5 a^2 b^3 x^{18} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {5 a^4 b x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {a^5 x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^5 \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 x^5 \left (a b+b^2 x\right )^5 \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (a^5 b^5 x^5+5 a^4 b^6 x^6+10 a^3 b^7 x^7+10 a^2 b^8 x^8+5 a b^9 x^9+b^{10} x^{10}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^5 x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 \left (a+b x^2\right )}+\frac {5 a^4 b x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {5 a^3 b^2 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {5 a^2 b^3 x^{18} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {b^5 x^{22} \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} \frac {x^{12} \sqrt {\left (a+b x^2\right )^2} \left (462 a^5+1980 a^4 b x^2+3465 a^3 b^2 x^4+3080 a^2 b^3 x^6+1386 a b^4 x^8+252 b^5 x^{10}\right )}{5544 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 22.75, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (462 a^5 x^{12}+1980 a^4 b x^{14}+3465 a^3 b^2 x^{16}+3080 a^2 b^3 x^{18}+1386 a b^4 x^{20}+252 b^5 x^{22}\right )}{5544 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{22} \, b^{5} x^{22} + \frac {1}{4} \, a b^{4} x^{20} + \frac {5}{9} \, a^{2} b^{3} x^{18} + \frac {5}{8} \, a^{3} b^{2} x^{16} + \frac {5}{14} \, a^{4} b x^{14} + \frac {1}{12} \, a^{5} x^{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 105, normalized size = 0.41 \begin {gather*} \frac {1}{22} \, b^{5} x^{22} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{4} \, a b^{4} x^{20} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{9} \, a^{2} b^{3} x^{18} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{8} \, a^{3} b^{2} x^{16} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{14} \, a^{4} b x^{14} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{12} \, a^{5} x^{12} \mathrm {sgn}\left (b x^{2} + 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 (252 b^{5} x^{10}+1386 a \,b^{4} x^{8}+3080 a^{2} b^{3} x^{6}+3465 a^{3} b^{2} x^{4}+1980 a^{4} b \,x^{2}+462 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{12}}{5544 \left (b \,x^{2}+a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{22} \, b^{5} x^{22} + \frac {1}{4} \, a b^{4} x^{20} + \frac {5}{9} \, a^{2} b^{3} x^{18} + \frac {5}{8} \, a^{3} b^{2} x^{16} + \frac {5}{14} \, a^{4} b x^{14} + \frac {1}{12} \, a^{5} x^{12} \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^{11}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\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^{11} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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