Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 x^{20} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 255, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 x^{20} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
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 )^{5/2}}{x^{21}} \, 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^{21}} \, dx}{b^4 \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^5 b^5}{x^{21}}+\frac {5 a^4 b^6}{x^{18}}+\frac {10 a^3 b^7}{x^{15}}+\frac {10 a^2 b^8}{x^{12}}+\frac {5 a b^9}{x^9}+\frac {b^{10}}{x^6}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{20 x^{20} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (2618 a^5+15400 a^4 b x^3+37400 a^3 b^2 x^6+47600 a^2 b^3 x^9+32725 a b^4 x^{12}+10472 b^5 x^{15}\right )}{52360 x^{20} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 59, normalized size = 0.23 \[ -\frac {10472 \, b^{5} x^{15} + 32725 \, a b^{4} x^{12} + 47600 \, a^{2} b^{3} x^{9} + 37400 \, a^{3} b^{2} x^{6} + 15400 \, a^{4} b x^{3} + 2618 \, a^{5}}{52360 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 107, normalized size = 0.42 \[ -\frac {10472 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 32725 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 47600 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 37400 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 15400 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2618 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{52360 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.31 \[ -\frac {\left (10472 b^{5} x^{15}+32725 a \,b^{4} x^{12}+47600 a^{2} b^{3} x^{9}+37400 a^{3} b^{2} x^{6}+15400 a^{4} b \,x^{3}+2618 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{52360 \left (b \,x^{3}+a \right )^{5} x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 59, normalized size = 0.23 \[ -\frac {10472 \, b^{5} x^{15} + 32725 \, a b^{4} x^{12} + 47600 \, a^{2} b^{3} x^{9} + 37400 \, a^{3} b^{2} x^{6} + 15400 \, a^{4} b x^{3} + 2618 \, a^{5}}{52360 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 231, normalized size = 0.91 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{20\,x^{20}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{5\,x^5\,\left (b\,x^3+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{8\,x^8\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{17\,x^{17}\,\left (b\,x^3+a\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{11\,x^{11}\,\left (b\,x^3+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^{14}\,\left (b\,x^3+a\right )} \]
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 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{21}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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