Optimal. Leaf size=38 \[ -\frac {2 a}{3 b^2 \sqrt {a+\frac {b}{x^3}}}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{3 b^2} \]
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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} \[ -\frac {2 a}{3 b^2 \sqrt {a+\frac {b}{x^3}}}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^7} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 a}{3 b^2 \sqrt {a+\frac {b}{x^3}}}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.76 \[ -\frac {2 \left (2 a x^3+b\right )}{3 b^2 x^3 \sqrt {a+\frac {b}{x^3}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 37, normalized size = 0.97 \[ -\frac {2 \, {\left (2 \, a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{3 \, {\left (a b^{2} x^{3} + b^{3}\right )}} \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.97 \[ -\frac {2 \left (a \,x^{3}+b \right ) \left (2 a \,x^{3}+b \right )}{3 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} b^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 30, normalized size = 0.79 \[ -\frac {2 \, \sqrt {a + \frac {b}{x^{3}}}}{3 \, b^{2}} - \frac {2 \, a}{3 \, \sqrt {a + \frac {b}{x^{3}}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 31, normalized size = 0.82 \[ -\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (2\,a\,x^3+b\right )}{3\,b^2\,\left (a\,x^3+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.15, size = 51, normalized size = 1.34 \[ \begin {cases} - \frac {4 a}{3 b^{2} \sqrt {a + \frac {b}{x^{3}}}} - \frac {2}{3 b x^{3} \sqrt {a + \frac {b}{x^{3}}}} & \text {for}\: b \neq 0 \\- \frac {1}{6 a^{\frac {3}{2}} x^{6}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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