Optimal. Leaf size=53 \[ -\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, 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 = {272, 45}
\begin {gather*} -\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{10}}-\frac {2 a}{b^2 (a+b x)^9}+\frac {1}{b^2 (a+b x)^8}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac {a}{8 b^3 \left (a+b x^2\right )^8}-\frac {1}{14 b^3 \left (a+b x^2\right )^7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.66 \begin {gather*} -\frac {a^2+9 a b x^2+36 b^2 x^4}{504 b^3 \left (a+b x^2\right )^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 48, normalized size = 0.91
method | result | size |
gosper | \(-\frac {36 b^{2} x^{4}+9 a b \,x^{2}+a^{2}}{504 \left (b \,x^{2}+a \right )^{9} b^{3}}\) | \(34\) |
norman | \(\frac {-\frac {a^{2}}{504 b^{3}}-\frac {a \,x^{2}}{56 b^{2}}-\frac {x^{4}}{14 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(37\) |
risch | \(\frac {-\frac {a^{2}}{504 b^{3}}-\frac {a \,x^{2}}{56 b^{2}}-\frac {x^{4}}{14 b}}{\left (b \,x^{2}+a \right )^{9}}\) | \(37\) |
default | \(-\frac {a^{2}}{18 b^{3} \left (b \,x^{2}+a \right )^{9}}+\frac {a}{8 b^{3} \left (b \,x^{2}+a \right )^{8}}-\frac {1}{14 b^{3} \left (b \,x^{2}+a \right )^{7}}\) | \(48\) |
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. 124 vs.
\(2 (47) = 94\).
time = 0.30, size = 124, normalized size = 2.34 \begin {gather*} -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} 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. 124 vs.
\(2 (47) = 94\).
time = 1.20, size = 124, normalized size = 2.34 \begin {gather*} -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} 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. 131 vs.
\(2 (46) = 92\).
time = 0.43, size = 131, normalized size = 2.47 \begin {gather*} \frac {- a^{2} - 9 a b x^{2} - 36 b^{2} x^{4}}{504 a^{9} b^{3} + 4536 a^{8} b^{4} x^{2} + 18144 a^{7} b^{5} x^{4} + 42336 a^{6} b^{6} x^{6} + 63504 a^{5} b^{7} x^{8} + 63504 a^{4} b^{8} x^{10} + 42336 a^{3} b^{9} x^{12} + 18144 a^{2} b^{10} x^{14} + 4536 a b^{11} x^{16} + 504 b^{12} x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 33, normalized size = 0.62 \begin {gather*} -\frac {36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.83, size = 125, normalized size = 2.36 \begin {gather*} -\frac {\frac {a^2}{504\,b^3}+\frac {x^4}{14\,b}+\frac {a\,x^2}{56\,b^2}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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