Optimal. Leaf size=167 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 167, 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 = {1355, 270} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^3}{x^{12}} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^3 b^3}{x^{12}}+\frac {3 a^2 b^4}{x^9}+\frac {3 a b^5}{x^6}+\frac {b^6}{x^3}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.37 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (40 a^3+165 a^2 b x^3+264 a b^2 x^6+220 b^3 x^9\right )}{440 x^{11} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 20.14, size = 61, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-40 a^3-165 a^2 b x^3-264 a b^2 x^6-220 b^3 x^9\right )}{440 x^{11} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 37, normalized size = 0.22 \begin {gather*} -\frac {220 \, b^{3} x^{9} + 264 \, a b^{2} x^{6} + 165 \, a^{2} b x^{3} + 40 \, a^{3}}{440 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 69, normalized size = 0.41 \begin {gather*} -\frac {220 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 264 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 165 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 40 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{440 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.35 \begin {gather*} -\frac {\left (220 b^{3} x^{9}+264 a \,b^{2} x^{6}+165 a^{2} b \,x^{3}+40 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{440 \left (b \,x^{3}+a \right )^{3} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 37, normalized size = 0.22 \begin {gather*} -\frac {220 \, b^{3} x^{9} + 264 \, a b^{2} x^{6} + 165 \, a^{2} b x^{3} + 40 \, a^{3}}{440 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 151, normalized size = 0.90 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{11\,x^{11}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{2\,x^2\,\left (b\,x^3+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{5\,x^5\,\left (b\,x^3+a\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{8\,x^8\,\left (b\,x^3+a\right )} \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 \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{12}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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