Optimal. Leaf size=41 \[ -\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}} \]
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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 )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}} \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 )^{5/2}}{x^{19}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{19}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {\left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 1.98 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a^5+6 a^4 b x^3+15 a^3 b^2 x^6+20 a^2 b^3 x^9+15 a b^4 x^{12}+6 b^5 x^{15}\right )}{18 x^{18} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.89, size = 442, normalized size = 10.78 \begin {gather*} \frac {16 b^5 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-a^{10} b-11 a^9 b^2 x^3-55 a^8 b^3 x^6-165 a^7 b^4 x^9-330 a^6 b^5 x^{12}-462 a^5 b^6 x^{15}-461 a^4 b^7 x^{18}-325 a^3 b^8 x^{21}-155 a^2 b^9 x^{24}-45 a b^{10} x^{27}-6 b^{11} x^{30}\right )+16 \sqrt {b^2} b^5 \left (a^{11}+12 a^{10} b x^3+66 a^9 b^2 x^6+220 a^8 b^3 x^9+495 a^7 b^4 x^{12}+792 a^6 b^5 x^{15}+923 a^5 b^6 x^{18}+786 a^4 b^7 x^{21}+480 a^3 b^8 x^{24}+200 a^2 b^9 x^{27}+51 a b^{10} x^{30}+6 b^{11} x^{33}\right )}{9 \sqrt {b^2} x^{18} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-32 a^5 b^5-160 a^4 b^6 x^3-320 a^3 b^7 x^6-320 a^2 b^8 x^9-160 a b^9 x^{12}-32 b^{10} x^{15}\right )+9 x^{18} \left (32 a^6 b^6+192 a^5 b^7 x^3+480 a^4 b^8 x^6+640 a^3 b^9 x^9+480 a^2 b^{10} x^{12}+192 a b^{11} x^{15}+32 b^{12} x^{18}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 57, normalized size = 1.39 \begin {gather*} -\frac {6 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 20 \, a^{2} b^{3} x^{9} + 15 \, a^{3} b^{2} x^{6} + 6 \, a^{4} b x^{3} + a^{5}}{18 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 106, normalized size = 2.59 \begin {gather*} -\frac {6 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 15 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 20 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 15 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 6 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{18 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 78, normalized size = 1.90 \begin {gather*} -\frac {\left (6 b^{5} x^{15}+15 a \,b^{4} x^{12}+20 a^{2} b^{3} x^{9}+15 a^{3} b^{2} x^{6}+6 a^{4} b \,x^{3}+a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{18 \left (b \,x^{3}+a \right )^{5} x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.10, size = 210, normalized size = 5.12 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{6}}{18 \, a^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{5}}{18 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{4}}{18 \, a^{6} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{3}}{18 \, a^{5} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{18 \, a^{4} x^{12}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{18 \, a^{3} x^{15}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{18 \, a^{2} x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 231, normalized size = 5.63 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{18\,x^{18}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^3\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{6\,x^6\,\left (b\,x^3+a\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^{15}\,\left (b\,x^3+a\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{9\,x^9\,\left (b\,x^3+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{6\,x^{12}\,\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 {5}{2}}}{x^{19}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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