Optimal. Leaf size=38 \[ \frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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} \[ \frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \[ \frac {2 \sqrt {a+\frac {b}{x^3}} \left (a x^3+b\right ) \left (2 a x^3-3 b\right )}{45 b^2 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 42, normalized size = 1.11 \[ \frac {2 \, {\left (2 \, a^{2} x^{6} - a b x^{3} - 3 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{45 \, b^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 29, normalized size = 0.76 \[ -\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} - 5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 1.03 \[ \frac {2 \left (a \,x^{3}+b \right ) \left (2 a \,x^{3}-3 b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{45 b^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 30, normalized size = 0.79 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b^{2}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 37, normalized size = 0.97 \[ -\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (-2\,a^2\,x^6+a\,b\,x^3+3\,b^2\right )}{45\,b^2\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.80, size = 313, normalized size = 8.24 \[ \frac {4 a^{\frac {11}{2}} b^{\frac {3}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {5}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {7}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {9}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {4 a^{6} b x^{\frac {21}{2}}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {4 a^{5} b^{2} x^{\frac {15}{2}}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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