Optimal. Leaf size=82 \[ -\frac {a^6}{15 x^{15}}-\frac {6 a^5 b}{13 x^{13}}-\frac {15 a^4 b^2}{11 x^{11}}-\frac {20 a^3 b^3}{9 x^9}-\frac {15 a^2 b^4}{7 x^7}-\frac {6 a b^5}{5 x^5}-\frac {b^6}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 82, 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}{15 x^{15}}-\frac {6 a^5 b}{13 x^{13}}-\frac {15 a^4 b^2}{11 x^{11}}-\frac {20 a^3 b^3}{9 x^9}-\frac {15 a^2 b^4}{7 x^7}-\frac {6 a b^5}{5 x^5}-\frac {b^6}{3 x^3} \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^{16}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{16}} \, dx}{b^6}\\ &=\frac {\int \left (\frac {a^6 b^6}{x^{16}}+\frac {6 a^5 b^7}{x^{14}}+\frac {15 a^4 b^8}{x^{12}}+\frac {20 a^3 b^9}{x^{10}}+\frac {15 a^2 b^{10}}{x^8}+\frac {6 a b^{11}}{x^6}+\frac {b^{12}}{x^4}\right ) \, dx}{b^6}\\ &=-\frac {a^6}{15 x^{15}}-\frac {6 a^5 b}{13 x^{13}}-\frac {15 a^4 b^2}{11 x^{11}}-\frac {20 a^3 b^3}{9 x^9}-\frac {15 a^2 b^4}{7 x^7}-\frac {6 a b^5}{5 x^5}-\frac {b^6}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 82, normalized size = 1.00 \begin {gather*} -\frac {a^6}{15 x^{15}}-\frac {6 a^5 b}{13 x^{13}}-\frac {15 a^4 b^2}{11 x^{11}}-\frac {20 a^3 b^3}{9 x^9}-\frac {15 a^2 b^4}{7 x^7}-\frac {6 a b^5}{5 x^5}-\frac {b^6}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 69, normalized size = 0.84
method | result | size |
default | \(-\frac {a^{6}}{15 x^{15}}-\frac {6 a^{5} b}{13 x^{13}}-\frac {15 a^{4} b^{2}}{11 x^{11}}-\frac {20 a^{3} b^{3}}{9 x^{9}}-\frac {15 a^{2} b^{4}}{7 x^{7}}-\frac {6 a \,b^{5}}{5 x^{5}}-\frac {b^{6}}{3 x^{3}}\) | \(69\) |
norman | \(\frac {-\frac {1}{15} a^{6}-\frac {6}{13} a^{5} b \,x^{2}-\frac {15}{11} a^{4} b^{2} x^{4}-\frac {20}{9} a^{3} b^{3} x^{6}-\frac {15}{7} a^{2} b^{4} x^{8}-\frac {6}{5} a \,b^{5} x^{10}-\frac {1}{3} b^{6} x^{12}}{x^{15}}\) | \(70\) |
risch | \(\frac {-\frac {1}{15} a^{6}-\frac {6}{13} a^{5} b \,x^{2}-\frac {15}{11} a^{4} b^{2} x^{4}-\frac {20}{9} a^{3} b^{3} x^{6}-\frac {15}{7} a^{2} b^{4} x^{8}-\frac {6}{5} a \,b^{5} x^{10}-\frac {1}{3} b^{6} x^{12}}{x^{15}}\) | \(70\) |
gosper | \(-\frac {15015 b^{6} x^{12}+54054 a \,b^{5} x^{10}+96525 a^{2} b^{4} x^{8}+100100 a^{3} b^{3} x^{6}+61425 a^{4} b^{2} x^{4}+20790 a^{5} b \,x^{2}+3003 a^{6}}{45045 x^{15}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 70, normalized size = 0.85 \begin {gather*} -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 70, normalized size = 0.85 \begin {gather*} -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 75, normalized size = 0.91 \begin {gather*} \frac {- 3003 a^{6} - 20790 a^{5} b x^{2} - 61425 a^{4} b^{2} x^{4} - 100100 a^{3} b^{3} x^{6} - 96525 a^{2} b^{4} x^{8} - 54054 a b^{5} x^{10} - 15015 b^{6} x^{12}}{45045 x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 70, normalized size = 0.85 \begin {gather*} -\frac {15015 \, b^{6} x^{12} + 54054 \, a b^{5} x^{10} + 96525 \, a^{2} b^{4} x^{8} + 100100 \, a^{3} b^{3} x^{6} + 61425 \, a^{4} b^{2} x^{4} + 20790 \, a^{5} b x^{2} + 3003 \, a^{6}}{45045 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 70, normalized size = 0.85 \begin {gather*} -\frac {\frac {a^6}{15}+\frac {6\,a^5\,b\,x^2}{13}+\frac {15\,a^4\,b^2\,x^4}{11}+\frac {20\,a^3\,b^3\,x^6}{9}+\frac {15\,a^2\,b^4\,x^8}{7}+\frac {6\,a\,b^5\,x^{10}}{5}+\frac {b^6\,x^{12}}{3}}{x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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