Optimal. Leaf size=116 \[ -\frac {1}{2 a^6 x^2}-\frac {b}{10 a^2 \left (a+b x^2\right )^5}-\frac {b}{4 a^3 \left (a+b x^2\right )^4}-\frac {b}{2 a^4 \left (a+b x^2\right )^3}-\frac {b}{a^5 \left (a+b x^2\right )^2}-\frac {5 b}{2 a^6 \left (a+b x^2\right )}-\frac {6 b \log (x)}{a^7}+\frac {3 b \log \left (a+b x^2\right )}{a^7} \]
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Rubi [A]
time = 0.09, antiderivative size = 116, 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, 46}
\begin {gather*} \frac {3 b \log \left (a+b x^2\right )}{a^7}-\frac {6 b \log (x)}{a^7}-\frac {5 b}{2 a^6 \left (a+b x^2\right )}-\frac {1}{2 a^6 x^2}-\frac {b}{a^5 \left (a+b x^2\right )^2}-\frac {b}{2 a^4 \left (a+b x^2\right )^3}-\frac {b}{4 a^3 \left (a+b x^2\right )^4}-\frac {b}{10 a^2 \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{x^3 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \text {Subst}\left (\int \frac {1}{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 {1}{a^6 b^6 x^2}-\frac {6}{a^7 b^5 x}+\frac {1}{a^2 b^4 (a+b x)^6}+\frac {2}{a^3 b^4 (a+b x)^5}+\frac {3}{a^4 b^4 (a+b x)^4}+\frac {4}{a^5 b^4 (a+b x)^3}+\frac {5}{a^6 b^4 (a+b x)^2}+\frac {6}{a^7 b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^6 x^2}-\frac {b}{10 a^2 \left (a+b x^2\right )^5}-\frac {b}{4 a^3 \left (a+b x^2\right )^4}-\frac {b}{2 a^4 \left (a+b x^2\right )^3}-\frac {b}{a^5 \left (a+b x^2\right )^2}-\frac {5 b}{2 a^6 \left (a+b x^2\right )}-\frac {6 b \log (x)}{a^7}+\frac {3 b \log \left (a+b x^2\right )}{a^7}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 92, normalized size = 0.79 \begin {gather*} -\frac {\frac {a \left (10 a^5+137 a^4 b x^2+385 a^3 b^2 x^4+470 a^2 b^3 x^6+270 a b^4 x^8+60 b^5 x^{10}\right )}{x^2 \left (a+b x^2\right )^5}+120 b \log (x)-60 b \log \left (a+b x^2\right )}{20 a^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 123, normalized size = 1.06
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {15 b^{2} x^{4}}{a^{3}}+\frac {45 b^{3} x^{6}}{a^{4}}+\frac {55 b^{4} x^{8}}{a^{5}}+\frac {125 b^{5} x^{10}}{4 a^{6}}+\frac {137 b^{6} x^{12}}{20 a^{7}}}{x^{2} \left (b \,x^{2}+a \right )^{5}}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {3 b \ln \left (b \,x^{2}+a \right )}{a^{7}}\) | \(98\) |
risch | \(\frac {-\frac {3 b^{5} x^{10}}{a^{6}}-\frac {27 b^{4} x^{8}}{2 a^{5}}-\frac {47 b^{3} x^{6}}{2 a^{4}}-\frac {77 b^{2} x^{4}}{4 a^{3}}-\frac {137 b \,x^{2}}{20 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}-\frac {6 b \ln \left (x \right )}{a^{7}}+\frac {3 b \ln \left (-b \,x^{2}-a \right )}{a^{7}}\) | \(119\) |
default | \(\frac {b^{2} \left (-\frac {a^{4}}{2 b \left (b \,x^{2}+a \right )^{4}}-\frac {5 a}{b \left (b \,x^{2}+a \right )}-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}-\frac {2 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {a^{3}}{b \left (b \,x^{2}+a \right )^{3}}+\frac {6 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{7}}-\frac {1}{2 a^{6} x^{2}}-\frac {6 b \ln \left (x \right )}{a^{7}}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 143, normalized size = 1.23 \begin {gather*} -\frac {60 \, b^{5} x^{10} + 270 \, a b^{4} x^{8} + 470 \, a^{2} b^{3} x^{6} + 385 \, a^{3} b^{2} x^{4} + 137 \, a^{4} b x^{2} + 10 \, a^{5}}{20 \, {\left (a^{6} b^{5} x^{12} + 5 \, a^{7} b^{4} x^{10} + 10 \, a^{8} b^{3} x^{8} + 10 \, a^{9} b^{2} x^{6} + 5 \, a^{10} b x^{4} + a^{11} x^{2}\right )}} + \frac {3 \, b \log \left (b x^{2} + a\right )}{a^{7}} - \frac {3 \, b \log \left (x^{2}\right )}{a^{7}} \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. 251 vs.
\(2 (106) = 212\).
time = 0.38, size = 251, normalized size = 2.16 \begin {gather*} -\frac {60 \, a b^{5} x^{10} + 270 \, a^{2} b^{4} x^{8} + 470 \, a^{3} b^{3} x^{6} + 385 \, a^{4} b^{2} x^{4} + 137 \, a^{5} b x^{2} + 10 \, a^{6} - 60 \, {\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (x\right )}{20 \, {\left (a^{7} b^{5} x^{12} + 5 \, a^{8} b^{4} x^{10} + 10 \, a^{9} b^{3} x^{8} + 10 \, a^{10} b^{2} x^{6} + 5 \, a^{11} b x^{4} + a^{12} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 150, normalized size = 1.29 \begin {gather*} \frac {- 10 a^{5} - 137 a^{4} b x^{2} - 385 a^{3} b^{2} x^{4} - 470 a^{2} b^{3} x^{6} - 270 a b^{4} x^{8} - 60 b^{5} x^{10}}{20 a^{11} x^{2} + 100 a^{10} b x^{4} + 200 a^{9} b^{2} x^{6} + 200 a^{8} b^{3} x^{8} + 100 a^{7} b^{4} x^{10} + 20 a^{6} b^{5} x^{12}} - \frac {6 b \log {\left (x \right )}}{a^{7}} + \frac {3 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.57, size = 115, normalized size = 0.99 \begin {gather*} -\frac {3 \, b \log \left (x^{2}\right )}{a^{7}} + \frac {3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{7}} + \frac {6 \, b x^{2} - a}{2 \, a^{7} x^{2}} - \frac {137 \, b^{6} x^{10} + 735 \, a b^{5} x^{8} + 1590 \, a^{2} b^{4} x^{6} + 1740 \, a^{3} b^{3} x^{4} + 970 \, a^{4} b^{2} x^{2} + 224 \, a^{5} b}{20 \, {\left (b x^{2} + a\right )}^{5} a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.68, size = 141, normalized size = 1.22 \begin {gather*} \frac {3\,b\,\ln \left (b\,x^2+a\right )}{a^7}-\frac {\frac {1}{2\,a}+\frac {137\,b\,x^2}{20\,a^2}+\frac {77\,b^2\,x^4}{4\,a^3}+\frac {47\,b^3\,x^6}{2\,a^4}+\frac {27\,b^4\,x^8}{2\,a^5}+\frac {3\,b^5\,x^{10}}{a^6}}{a^5\,x^2+5\,a^4\,b\,x^4+10\,a^3\,b^2\,x^6+10\,a^2\,b^3\,x^8+5\,a\,b^4\,x^{10}+b^5\,x^{12}}-\frac {6\,b\,\ln \left (x\right )}{a^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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