Optimal. Leaf size=34 \[ \frac {a}{10 b^2 \left (a+b x^2\right )^5}-\frac {1}{8 b^2 \left (a+b x^2\right )^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45}
\begin {gather*} \frac {a}{10 b^2 \left (a+b x^2\right )^5}-\frac {1}{8 b^2 \left (a+b x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^3}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \frac {x}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \left (-\frac {a}{b^7 (a+b x)^6}+\frac {1}{b^7 (a+b x)^5}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{10 b^2 \left (a+b x^2\right )^5}-\frac {1}{8 b^2 \left (a+b x^2\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} -\frac {a+5 b x^2}{40 b^2 \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 31, normalized size = 0.91
method | result | size |
norman | \(\frac {-\frac {x^{2}}{8 b}-\frac {a}{40 b^{2}}}{\left (b \,x^{2}+a \right )^{5}}\) | \(26\) |
default | \(\frac {a}{10 b^{2} \left (b \,x^{2}+a \right )^{5}}-\frac {1}{8 b^{2} \left (b \,x^{2}+a \right )^{4}}\) | \(31\) |
gosper | \(-\frac {5 b \,x^{2}+a}{40 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} b^{2}}\) | \(43\) |
risch | \(\frac {-\frac {x^{2}}{8 b}-\frac {a}{40 b^{2}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}\) | \(46\) |
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. 69 vs.
\(2 (30) = 60\).
time = 0.28, size = 69, normalized size = 2.03 \begin {gather*} -\frac {5 \, b x^{2} + a}{40 \, {\left (b^{7} x^{10} + 5 \, a b^{6} x^{8} + 10 \, a^{2} b^{5} x^{6} + 10 \, a^{3} b^{4} x^{4} + 5 \, a^{4} b^{3} x^{2} + a^{5} b^{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. 69 vs.
\(2 (30) = 60\).
time = 0.32, size = 69, normalized size = 2.03 \begin {gather*} -\frac {5 \, b x^{2} + a}{40 \, {\left (b^{7} x^{10} + 5 \, a b^{6} x^{8} + 10 \, a^{2} b^{5} x^{6} + 10 \, a^{3} b^{4} x^{4} + 5 \, a^{4} b^{3} x^{2} + a^{5} b^{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. 71 vs.
\(2 (29) = 58\).
time = 0.25, size = 71, normalized size = 2.09 \begin {gather*} \frac {- a - 5 b x^{2}}{40 a^{5} b^{2} + 200 a^{4} b^{3} x^{2} + 400 a^{3} b^{4} x^{4} + 400 a^{2} b^{5} x^{6} + 200 a b^{6} x^{8} + 40 b^{7} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.79, size = 22, normalized size = 0.65 \begin {gather*} -\frac {5 \, b x^{2} + a}{40 \, {\left (b x^{2} + a\right )}^{5} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.48, size = 70, normalized size = 2.06 \begin {gather*} -\frac {\frac {a}{40\,b^2}+\frac {x^2}{8\,b}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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