Optimal. Leaf size=43 \[ -\frac {b^3}{9 x^9}-\frac {3 a b^2}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{3 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 276}
\begin {gather*} -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 269
Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^4} \, dx &=\int \frac {\left (b+a x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac {b^3}{x^{10}}+\frac {3 a b^2}{x^8}+\frac {3 a^2 b}{x^6}+\frac {a^3}{x^4}\right ) \, dx\\ &=-\frac {b^3}{9 x^9}-\frac {3 a b^2}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {b^3}{9 x^9}-\frac {3 a b^2}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 36, normalized size = 0.84
method | result | size |
default | \(-\frac {b^{3}}{9 x^{9}}-\frac {3 a \,b^{2}}{7 x^{7}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a^{3}}{3 x^{3}}\) | \(36\) |
norman | \(\frac {-\frac {1}{3} a^{3} x^{6}-\frac {3}{5} a^{2} b \,x^{4}-\frac {3}{7} a \,b^{2} x^{2}-\frac {1}{9} b^{3}}{x^{9}}\) | \(37\) |
risch | \(\frac {-\frac {1}{3} a^{3} x^{6}-\frac {3}{5} a^{2} b \,x^{4}-\frac {3}{7} a \,b^{2} x^{2}-\frac {1}{9} b^{3}}{x^{9}}\) | \(37\) |
gosper | \(-\frac {105 a^{3} x^{6}+189 a^{2} b \,x^{4}+135 a \,b^{2} x^{2}+35 b^{3}}{315 x^{9}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 39, normalized size = 0.91 \begin {gather*} \frac {- 105 a^{3} x^{6} - 189 a^{2} b x^{4} - 135 a b^{2} x^{2} - 35 b^{3}}{315 x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 37, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^3\,x^6}{3}+\frac {3\,a^2\,b\,x^4}{5}+\frac {3\,a\,b^2\,x^2}{7}+\frac {b^3}{9}}{x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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