Optimal. Leaf size=59 \[ -\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{10}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{9/2}}{27 b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.86 \[ -\frac {2 \sqrt {a+\frac {b}{x^3}} \left (a x^3+b\right )^2 \left (8 a^2 x^6-20 a b x^3+35 b^2\right )}{945 b^3 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 64, normalized size = 1.08 \[ -\frac {2 \, {\left (8 \, a^{4} x^{12} - 4 \, a^{3} b x^{9} + 3 \, a^{2} b^{2} x^{6} + 50 \, a b^{3} x^{3} + 35 \, b^{4}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{945 \, b^{3} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 43, normalized size = 0.73 \[ -\frac {2 \, {\left (35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}} - 90 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a + 63 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{2}\right )}}{945 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.85 \[ -\frac {2 \left (a \,x^{3}+b \right ) \left (8 a^{2} x^{6}-20 a b \,x^{3}+35 b^{2}\right ) \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}{945 b^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}}}{27 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a}{21 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{2}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.85, size = 88, normalized size = 1.49 \[ \frac {8\,a^3\,\sqrt {a+\frac {b}{x^3}}}{945\,b^2\,x^3}-\frac {2\,b\,\sqrt {a+\frac {b}{x^3}}}{27\,x^{12}}-\frac {16\,a^4\,\sqrt {a+\frac {b}{x^3}}}{945\,b^3}-\frac {20\,a\,\sqrt {a+\frac {b}{x^3}}}{189\,x^9}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x^3}}}{315\,b\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.24, size = 1001, normalized size = 16.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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