Optimal. Leaf size=41 \[ -\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 264} \begin {gather*} -\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{13}} \, 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^{13}} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a^3+4 a^2 b x^3+6 a b^2 x^6+4 b^3 x^9\right )}{12 x^{12} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.35, size = 310, normalized size = 7.56 \begin {gather*} \frac {2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-a^6 b-7 a^5 b^2 x^3-21 a^4 b^3 x^6-35 a^3 b^4 x^9-34 a^2 b^5 x^{12}-18 a b^6 x^{15}-4 b^7 x^{18}\right )+2 \sqrt {b^2} b^3 \left (a^7+8 a^6 b x^3+28 a^5 b^2 x^6+56 a^4 b^3 x^9+69 a^3 b^4 x^{12}+52 a^2 b^5 x^{15}+22 a b^6 x^{18}+4 b^7 x^{21}\right )}{3 \sqrt {b^2} x^{12} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-8 a^3 b^3-24 a^2 b^4 x^3-24 a b^5 x^6-8 b^6 x^9\right )+3 x^{12} \left (8 a^4 b^4+32 a^3 b^5 x^3+48 a^2 b^6 x^6+32 a b^7 x^9+8 b^8 x^{12}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 35, normalized size = 0.85 \begin {gather*} -\frac {4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 68, normalized size = 1.66 \begin {gather*} -\frac {4 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 6 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{12 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 1.37 \begin {gather*} -\frac {\left (4 b^{3} x^{9}+6 a \,b^{2} x^{6}+4 a^{2} b \,x^{3}+a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{12 \left (b \,x^{3}+a \right )^{3} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 148, normalized size = 3.61 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{4}}{12 \, a^{4}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{3}}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{2}}{12 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b}{12 \, a^{3} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}}}{12 \, a^{2} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 151, normalized size = 3.68 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{12\,x^{12}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^3\,\left (b\,x^3+a\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{2\,x^6\,\left (b\,x^3+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^9\,\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^{13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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