Optimal. Leaf size=251 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 251, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^5}{x^{12}} \, 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 (\frac {a^5 b^5}{x^{12}}+\frac {5 a^4 b^6}{x^{10}}+\frac {10 a^3 b^7}{x^8}+\frac {10 a^2 b^8}{x^6}+\frac {5 a b^9}{x^4}+\frac {b^{10}}{x^2}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5 \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (63 a^5+385 a^4 b x^2+990 a^3 b^2 x^4+1386 a^2 b^3 x^6+1155 a b^4 x^8+693 b^5 x^{10}\right )}{693 x^{11} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 59, normalized size = 0.24 \[ -\frac {693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 107, normalized size = 0.43 \[ -\frac {693 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 1155 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 1386 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 990 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 385 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 63 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{693 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.32 \[ -\frac {\left (693 b^{5} x^{10}+1155 a \,b^{4} x^{8}+1386 a^{2} b^{3} x^{6}+990 a^{3} b^{2} x^{4}+385 a^{4} b \,x^{2}+63 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{693 \left (b \,x^{2}+a \right )^{5} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 57, normalized size = 0.23 \[ -\frac {b^{5}}{x} - \frac {5 \, a b^{4}}{3 \, x^{3}} - \frac {2 \, a^{2} b^{3}}{x^{5}} - \frac {10 \, a^{3} b^{2}}{7 \, x^{7}} - \frac {5 \, a^{4} b}{9 \, x^{9}} - \frac {a^{5}}{11 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 231, normalized size = 0.92 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{11\,x^{11}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{3\,x^3\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{9\,x^9\,\left (b\,x^2+a\right )}-\frac {2\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^5\,\left (b\,x^2+a\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{7\,x^7\,\left (b\,x^2+a\right )} \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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