Optimal. Leaf size=80 \[ -\frac {a^6}{19 x^{19}}-\frac {6 a^5 b}{17 x^{17}}-\frac {a^4 b^2}{x^{15}}-\frac {20 a^3 b^3}{13 x^{13}}-\frac {15 a^2 b^4}{11 x^{11}}-\frac {2 a b^5}{3 x^9}-\frac {b^6}{7 x^7} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276}
\begin {gather*} -\frac {a^6}{19 x^{19}}-\frac {6 a^5 b}{17 x^{17}}-\frac {a^4 b^2}{x^{15}}-\frac {20 a^3 b^3}{13 x^{13}}-\frac {15 a^2 b^4}{11 x^{11}}-\frac {2 a b^5}{3 x^9}-\frac {b^6}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{20}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{20}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{x^{20}}+\frac {6 a^5 b^7}{x^{18}}+\frac {15 a^4 b^8}{x^{16}}+\frac {20 a^3 b^9}{x^{14}}+\frac {15 a^2 b^{10}}{x^{12}}+\frac {6 a b^{11}}{x^{10}}+\frac {b^{12}}{x^8}\right ) \, dx}{b^6}\\ &=-\frac {a^6}{19 x^{19}}-\frac {6 a^5 b}{17 x^{17}}-\frac {a^4 b^2}{x^{15}}-\frac {20 a^3 b^3}{13 x^{13}}-\frac {15 a^2 b^4}{11 x^{11}}-\frac {2 a b^5}{3 x^9}-\frac {b^6}{7 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 80, normalized size = 1.00 \begin {gather*} -\frac {a^6}{19 x^{19}}-\frac {6 a^5 b}{17 x^{17}}-\frac {a^4 b^2}{x^{15}}-\frac {20 a^3 b^3}{13 x^{13}}-\frac {15 a^2 b^4}{11 x^{11}}-\frac {2 a b^5}{3 x^9}-\frac {b^6}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 69, normalized size = 0.86
method | result | size |
default | \(-\frac {a^{6}}{19 x^{19}}-\frac {6 a^{5} b}{17 x^{17}}-\frac {a^{4} b^{2}}{x^{15}}-\frac {20 a^{3} b^{3}}{13 x^{13}}-\frac {15 a^{2} b^{4}}{11 x^{11}}-\frac {2 a \,b^{5}}{3 x^{9}}-\frac {b^{6}}{7 x^{7}}\) | \(69\) |
norman | \(\frac {-\frac {1}{19} a^{6}-\frac {6}{17} a^{5} b \,x^{2}-a^{4} b^{2} x^{4}-\frac {20}{13} a^{3} b^{3} x^{6}-\frac {15}{11} a^{2} b^{4} x^{8}-\frac {2}{3} a \,b^{5} x^{10}-\frac {1}{7} b^{6} x^{12}}{x^{19}}\) | \(70\) |
risch | \(\frac {-\frac {1}{19} a^{6}-\frac {6}{17} a^{5} b \,x^{2}-a^{4} b^{2} x^{4}-\frac {20}{13} a^{3} b^{3} x^{6}-\frac {15}{11} a^{2} b^{4} x^{8}-\frac {2}{3} a \,b^{5} x^{10}-\frac {1}{7} b^{6} x^{12}}{x^{19}}\) | \(70\) |
gosper | \(-\frac {138567 b^{6} x^{12}+646646 a \,b^{5} x^{10}+1322685 a^{2} b^{4} x^{8}+1492260 a^{3} b^{3} x^{6}+969969 a^{4} b^{2} x^{4}+342342 a^{5} b \,x^{2}+51051 a^{6}}{969969 x^{19}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 70, normalized size = 0.88 \begin {gather*} -\frac {138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 70, normalized size = 0.88 \begin {gather*} -\frac {138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 75, normalized size = 0.94 \begin {gather*} \frac {- 51051 a^{6} - 342342 a^{5} b x^{2} - 969969 a^{4} b^{2} x^{4} - 1492260 a^{3} b^{3} x^{6} - 1322685 a^{2} b^{4} x^{8} - 646646 a b^{5} x^{10} - 138567 b^{6} x^{12}}{969969 x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.25, size = 70, normalized size = 0.88 \begin {gather*} -\frac {138567 \, b^{6} x^{12} + 646646 \, a b^{5} x^{10} + 1322685 \, a^{2} b^{4} x^{8} + 1492260 \, a^{3} b^{3} x^{6} + 969969 \, a^{4} b^{2} x^{4} + 342342 \, a^{5} b x^{2} + 51051 \, a^{6}}{969969 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 69, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^6}{19}+\frac {6\,a^5\,b\,x^2}{17}+a^4\,b^2\,x^4+\frac {20\,a^3\,b^3\,x^6}{13}+\frac {15\,a^2\,b^4\,x^8}{11}+\frac {2\,a\,b^5\,x^{10}}{3}+\frac {b^6\,x^{12}}{7}}{x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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