Optimal. Leaf size=38 \[ \frac {2 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^2} \]
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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} \[ \frac {2 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^2} \]
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^7} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int 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 (a+b x)^{3/2}}{b}+\frac {(a+b x)^{5/2}}{b}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=\frac {2 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.05 \[ \frac {2 \sqrt {a+\frac {b}{x^3}} \left (a x^3+b\right )^2 \left (2 a x^3-5 b\right )}{105 b^2 x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 53, normalized size = 1.39 \[ \frac {2 \, {\left (2 \, a^{3} x^{9} - a^{2} b x^{6} - 8 \, a b^{2} x^{3} - 5 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{105 \, b^{2} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 29, normalized size = 0.76 \[ -\frac {2 \, {\left (5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 7 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a\right )}}{105 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 1.03 \[ \frac {2 \left (a \,x^{3}+b \right ) \left (2 a \,x^{3}-5 b \right ) \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}{105 b^{2} x^{6}} \]
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 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}}}{21 \, b^{2}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 68, normalized size = 1.79 \[ \frac {4\,a^3\,\sqrt {a+\frac {b}{x^3}}}{105\,b^2}-\frac {2\,b\,\sqrt {a+\frac {b}{x^3}}}{21\,x^9}-\frac {16\,a\,\sqrt {a+\frac {b}{x^3}}}{105\,x^6}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x^3}}}{105\,b\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.05, size = 371, normalized size = 9.76 \[ \frac {4 a^{\frac {15}{2}} b^{\frac {3}{2}} x^{12} \sqrt {\frac {a x^{3}}{b} + 1}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} + \frac {2 a^{\frac {13}{2}} b^{\frac {5}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} - \frac {18 a^{\frac {11}{2}} b^{\frac {7}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} - \frac {26 a^{\frac {9}{2}} b^{\frac {9}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} - \frac {10 a^{\frac {7}{2}} b^{\frac {11}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} - \frac {4 a^{8} b x^{\frac {27}{2}}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} - \frac {4 a^{7} b^{2} x^{\frac {21}{2}}}{105 a^{\frac {9}{2}} b^{3} x^{\frac {27}{2}} + 105 a^{\frac {7}{2}} b^{4} x^{\frac {21}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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