Optimal. Leaf size=248 \[ \frac {b^5 x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{11 \left (a+b x^2\right )^5}+\frac {5 a b^4 x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{9 \left (a+b x^2\right )^5}+\frac {10 a^2 b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{7 \left (a+b x^2\right )^5}+\frac {a^5 x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac {5 a^4 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{3 \left (a+b x^2\right )^5}+\frac {2 a^3 b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.05, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1088, 194} \[ \frac {b^5 x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{11 \left (a+b x^2\right )^5}+\frac {5 a b^4 x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{9 \left (a+b x^2\right )^5}+\frac {10 a^2 b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{7 \left (a+b x^2\right )^5}+\frac {2 a^3 b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac {5 a^4 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{3 \left (a+b x^2\right )^5}+\frac {a^5 x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 194
Rule 1088
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \int \left (2 a b+2 b^2 x^2\right )^5 \, dx}{\left (2 a b+2 b^2 x^2\right )^5}\\ &=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \int \left (32 a^5 b^5+160 a^4 b^6 x^2+320 a^3 b^7 x^4+320 a^2 b^8 x^6+160 a b^9 x^8+32 b^{10} x^{10}\right ) \, dx}{\left (2 a b+2 b^2 x^2\right )^5}\\ &=\frac {a^5 x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac {5 a^4 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{3 \left (a+b x^2\right )^5}+\frac {2 a^3 b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac {10 a^2 b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{7 \left (a+b x^2\right )^5}+\frac {5 a b^4 x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{9 \left (a+b x^2\right )^5}+\frac {b^5 x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{11 \left (a+b x^2\right )^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 0.33 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (693 a^5 x+1155 a^4 b x^3+1386 a^3 b^2 x^5+990 a^2 b^3 x^7+385 a b^4 x^9+63 b^5 x^{11}\right )}{693 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 54, normalized size = 0.22 \[ \frac {1}{11} \, b^{5} x^{11} + \frac {5}{9} \, a b^{4} x^{9} + \frac {10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{3} \, a^{4} b x^{3} + a^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 102, normalized size = 0.41 \[ \frac {1}{11} \, b^{5} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{9} \, a b^{4} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{7} \, a^{2} b^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{3} \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{5} x \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 78, normalized size = 0.31 \[ \frac {\left (63 b^{5} x^{10}+385 a \,b^{4} x^{8}+990 a^{2} b^{3} x^{6}+1386 a^{3} b^{2} x^{4}+1155 a^{4} b \,x^{2}+693 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x}{693 \left (b \,x^{2}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 54, normalized size = 0.22 \[ \frac {1}{11} \, b^{5} x^{11} + \frac {5}{9} \, a b^{4} x^{9} + \frac {10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{3} \, a^{4} b x^{3} + a^{5} x \]
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 \[ \int {\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]
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 \[ \int \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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