Optimal. Leaf size=19 \[ \frac {x^4}{4 b \left (b+c x^2\right )^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 270}
\begin {gather*} \frac {x^4}{4 b \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^3}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {x^4}{4 b \left (b+c x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {b+2 c x^2}{4 c^2 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 31, normalized size = 1.63
method | result | size |
gosper | \(-\frac {2 c \,x^{2}+b}{4 \left (c \,x^{2}+b \right )^{2} c^{2}}\) | \(23\) |
risch | \(\frac {-\frac {x^{2}}{2 c}-\frac {b}{4 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}\) | \(26\) |
default | \(-\frac {1}{2 c^{2} \left (c \,x^{2}+b \right )}+\frac {b}{4 c^{2} \left (c \,x^{2}+b \right )^{2}}\) | \(31\) |
norman | \(\frac {-\frac {x^{7}}{2 c}-\frac {b \,x^{5}}{4 c^{2}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}\) | \(32\) |
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. 36 vs.
\(2 (17) = 34\).
time = 0.30, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, c x^{2} + b}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\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. 36 vs.
\(2 (17) = 34\).
time = 0.36, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, c x^{2} + b}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\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. 36 vs.
\(2 (14) = 28\).
time = 0.13, size = 36, normalized size = 1.89 \begin {gather*} \frac {- b - 2 c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.83, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, c x^{2} + b}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.18, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {b}{4\,c^2}+\frac {x^2}{2\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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