Optimal. Leaf size=19 \[ \frac {x^6}{6 a \left (a+b x^2\right )^3} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270}
\begin {gather*} \frac {x^6}{6 a \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^5}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {x^6}{6 a \left (a+b x^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.84 \begin {gather*} -\frac {a^2+3 a b x^2+3 b^2 x^4}{6 b^3 \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs.
\(2(17)=34\).
time = 0.03, size = 48, normalized size = 2.53
method | result | size |
norman | \(\frac {-\frac {x^{4}}{2 b}-\frac {a \,x^{2}}{2 b^{2}}-\frac {a^{2}}{6 b^{3}}}{\left (b \,x^{2}+a \right )^{3}}\) | \(37\) |
default | \(\frac {a}{2 b^{3} \left (b \,x^{2}+a \right )^{2}}-\frac {1}{2 b^{3} \left (b \,x^{2}+a \right )}-\frac {a^{2}}{6 b^{3} \left (b \,x^{2}+a \right )^{3}}\) | \(48\) |
gosper | \(-\frac {3 b^{2} x^{4}+3 a b \,x^{2}+a^{2}}{6 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) b^{3}}\) | \(54\) |
risch | \(\frac {-\frac {x^{4}}{2 b}-\frac {a \,x^{2}}{2 b^{2}}-\frac {a^{2}}{6 b^{3}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (17) = 34\).
time = 0.33, size = 58, normalized size = 3.05 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (17) = 34\).
time = 0.34, size = 58, normalized size = 3.05 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (14) = 28\).
time = 0.17, size = 60, normalized size = 3.16 \begin {gather*} \frac {- a^{2} - 3 a b x^{2} - 3 b^{2} x^{4}}{6 a^{3} b^{3} + 18 a^{2} b^{4} x^{2} + 18 a b^{5} x^{4} + 6 b^{6} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.02, size = 33, normalized size = 1.74 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, {\left (b x^{2} + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.29, size = 60, normalized size = 3.16 \begin {gather*} -\frac {a^2+3\,a\,b\,x^2+3\,b^2\,x^4}{6\,a^3\,b^3+18\,a^2\,b^4\,x^2+18\,a\,b^5\,x^4+6\,b^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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