Optimal. Leaf size=53 \[ -\frac {a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, 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^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^5}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \frac {x^2}{\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^2}{b^8 (a+b x)^6}-\frac {2 a}{b^8 (a+b x)^5}+\frac {1}{b^8 (a+b x)^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.66 \begin {gather*} -\frac {a^2+5 a b x^2+10 b^2 x^4}{60 b^3 \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 48, normalized size = 0.91
method | result | size |
norman | \(\frac {-\frac {x^{4}}{6 b}-\frac {a \,x^{2}}{12 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b \,x^{2}+a \right )^{5}}\) | \(37\) |
default | \(-\frac {a^{2}}{10 b^{3} \left (b \,x^{2}+a \right )^{5}}+\frac {a}{4 b^{3} \left (b \,x^{2}+a \right )^{4}}-\frac {1}{6 b^{3} \left (b \,x^{2}+a \right )^{3}}\) | \(48\) |
gosper | \(-\frac {10 b^{2} x^{4}+5 a b \,x^{2}+a^{2}}{60 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} b^{3}}\) | \(54\) |
risch | \(\frac {-\frac {x^{4}}{6 b}-\frac {a \,x^{2}}{12 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 80, normalized size = 1.51 \begin {gather*} -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 80, normalized size = 1.51 \begin {gather*} -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 83, normalized size = 1.57 \begin {gather*} \frac {- a^{2} - 5 a b x^{2} - 10 b^{2} x^{4}}{60 a^{5} b^{3} + 300 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{4} + 600 a^{2} b^{6} x^{6} + 300 a b^{7} x^{8} + 60 b^{8} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.11, size = 33, normalized size = 0.62 \begin {gather*} -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b x^{2} + a\right )}^{5} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.62, size = 81, normalized size = 1.53 \begin {gather*} -\frac {\frac {a^2}{60\,b^3}+\frac {x^4}{6\,b}+\frac {a\,x^2}{12\,b^2}}{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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