Optimal. Leaf size=59 \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{3 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3} \]
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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} \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{3 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{10}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 \sqrt {a+b x}}-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {(a+b x)^{3/2}}{b^2}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 a^2 \sqrt {a+\frac {b}{x^3}}}{3 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.71 \[ -\frac {2 \sqrt {a+\frac {b}{x^3}} \left (8 a^2 x^6-4 a b x^3+3 b^2\right )}{45 b^3 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 42, normalized size = 0.71 \[ -\frac {2 \, {\left (8 \, a^{2} x^{6} - 4 \, a b x^{3} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{45 \, b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 0.85 \[ -\frac {2 \left (a \,x^{3}+b \right ) \left (8 a^{2} x^{6}-4 a b \,x^{3}+3 b^{2}\right )}{45 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, b^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a}{9 \, b^{3}} - \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{2}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 56, normalized size = 0.95 \[ -\frac {6\,b^2\,\sqrt {a+\frac {b}{x^3}}+16\,a^2\,x^6\,\sqrt {a+\frac {b}{x^3}}-8\,a\,b\,x^3\,\sqrt {a+\frac {b}{x^3}}}{45\,b^3\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.19, size = 824, normalized size = 13.97 \[ - \frac {16 a^{\frac {15}{2}} b^{\frac {9}{2}} x^{15} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {11}{2}} x^{12} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} - \frac {30 a^{\frac {11}{2}} b^{\frac {13}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} - \frac {10 a^{\frac {9}{2}} b^{\frac {15}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} - \frac {10 a^{\frac {7}{2}} b^{\frac {17}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {19}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} + \frac {16 a^{8} b^{4} x^{\frac {33}{2}}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} + \frac {48 a^{7} b^{5} x^{\frac {27}{2}}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} + \frac {48 a^{6} b^{6} x^{\frac {21}{2}}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} + \frac {16 a^{5} b^{7} x^{\frac {15}{2}}}{45 a^{\frac {11}{2}} b^{7} x^{\frac {33}{2}} + 135 a^{\frac {9}{2}} b^{8} x^{\frac {27}{2}} + 135 a^{\frac {7}{2}} b^{9} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{10} x^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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