Optimal. Leaf size=54 \[ \frac {a^2}{b^3 \sqrt {a+\frac {b}{x^2}}}+\frac {2 a \sqrt {a+\frac {b}{x^2}}}{b^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 54, 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} \[ \frac {a^2}{b^3 \sqrt {a+\frac {b}{x^2}}}+\frac {2 a \sqrt {a+\frac {b}{x^2}}}{b^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{3/2}}-\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b^2}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a^2}{b^3 \sqrt {a+\frac {b}{x^2}}}+\frac {2 a \sqrt {a+\frac {b}{x^2}}}{b^3}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.78 \[ \frac {8 a^2 x^4+4 a b x^2-b^2}{3 b^3 x^4 \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 54, normalized size = 1.00 \[ \frac {{\left (8 \, a^{2} x^{4} + 4 \, a b x^{2} - b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]
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 \[ \int \frac {1}{{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.93 \[ \frac {\left (a \,x^{2}+b \right ) \left (8 a^{2} x^{4}+4 a b \,x^{2}-b^{2}\right )}{3 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 46, normalized size = 0.85 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {2 \, \sqrt {a + \frac {b}{x^{2}}} a}{b^{3}} + \frac {a^{2}}{\sqrt {a + \frac {b}{x^{2}}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 47, normalized size = 0.87 \[ \frac {\sqrt {a+\frac {b}{x^2}}\,\left (8\,a^2\,x^4+4\,a\,b\,x^2-b^2\right )}{3\,b^3\,x^2\,\left (a\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.60, size = 423, normalized size = 7.83 \[ \frac {8 a^{\frac {9}{2}} b^{\frac {7}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} + \frac {12 a^{\frac {7}{2}} b^{\frac {9}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} + \frac {3 a^{\frac {5}{2}} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} - \frac {a^{\frac {3}{2}} b^{\frac {13}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} - \frac {8 a^{5} b^{3} x^{7}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} - \frac {16 a^{4} b^{4} x^{5}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} - \frac {8 a^{3} b^{5} x^{3}}{3 a^{\frac {7}{2}} b^{6} x^{7} + 6 a^{\frac {5}{2}} b^{7} x^{5} + 3 a^{\frac {3}{2}} b^{8} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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