Integrand size = 24, antiderivative size = 77 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=-\frac {a^6}{2 x^2}+\frac {15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac {5}{2} a^2 b^4 x^6+\frac {3}{4} a b^5 x^8+\frac {b^6 x^{10}}{10}+6 a^5 b \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=-\frac {a^6}{2 x^2}+6 a^5 b \log (x)+\frac {15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac {5}{2} a^2 b^4 x^6+\frac {3}{4} a b^5 x^8+\frac {b^6 x^{10}}{10} \]
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Rule 28
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^3} \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^2} \, dx,x,x^2\right )}{2 b^6} \\ & = \frac {\text {Subst}\left (\int \left (15 a^4 b^8+\frac {a^6 b^6}{x^2}+\frac {6 a^5 b^7}{x}+20 a^3 b^9 x+15 a^2 b^{10} x^2+6 a b^{11} x^3+b^{12} x^4\right ) \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {a^6}{2 x^2}+\frac {15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac {5}{2} a^2 b^4 x^6+\frac {3}{4} a b^5 x^8+\frac {b^6 x^{10}}{10}+6 a^5 b \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=-\frac {a^6}{2 x^2}+\frac {15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac {5}{2} a^2 b^4 x^6+\frac {3}{4} a b^5 x^8+\frac {b^6 x^{10}}{10}+6 a^5 b \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{6}}{2 x^{2}}+\frac {15 a^{4} b^{2} x^{2}}{2}+5 a^{3} b^{3} x^{4}+\frac {5 a^{2} b^{4} x^{6}}{2}+\frac {3 a \,b^{5} x^{8}}{4}+\frac {b^{6} x^{10}}{10}+6 a^{5} b \ln \left (x \right )\) | \(68\) |
risch | \(-\frac {a^{6}}{2 x^{2}}+\frac {15 a^{4} b^{2} x^{2}}{2}+5 a^{3} b^{3} x^{4}+\frac {5 a^{2} b^{4} x^{6}}{2}+\frac {3 a \,b^{5} x^{8}}{4}+\frac {b^{6} x^{10}}{10}+6 a^{5} b \ln \left (x \right )\) | \(68\) |
norman | \(\frac {-\frac {1}{2} a^{6}+\frac {1}{10} b^{6} x^{12}+\frac {3}{4} a \,b^{5} x^{10}+\frac {5}{2} a^{2} b^{4} x^{8}+5 a^{3} b^{3} x^{6}+\frac {15}{2} a^{4} b^{2} x^{4}}{x^{2}}+6 a^{5} b \ln \left (x \right )\) | \(70\) |
parallelrisch | \(\frac {2 b^{6} x^{12}+15 a \,b^{5} x^{10}+50 a^{2} b^{4} x^{8}+100 a^{3} b^{3} x^{6}+150 a^{4} b^{2} x^{4}+120 a^{5} b \ln \left (x \right ) x^{2}-10 a^{6}}{20 x^{2}}\) | \(73\) |
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Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=\frac {2 \, b^{6} x^{12} + 15 \, a b^{5} x^{10} + 50 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 150 \, a^{4} b^{2} x^{4} + 120 \, a^{5} b x^{2} \log \left (x\right ) - 10 \, a^{6}}{20 \, x^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=- \frac {a^{6}}{2 x^{2}} + 6 a^{5} b \log {\left (x \right )} + \frac {15 a^{4} b^{2} x^{2}}{2} + 5 a^{3} b^{3} x^{4} + \frac {5 a^{2} b^{4} x^{6}}{2} + \frac {3 a b^{5} x^{8}}{4} + \frac {b^{6} x^{10}}{10} \]
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Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=\frac {1}{10} \, b^{6} x^{10} + \frac {3}{4} \, a b^{5} x^{8} + \frac {5}{2} \, a^{2} b^{4} x^{6} + 5 \, a^{3} b^{3} x^{4} + \frac {15}{2} \, a^{4} b^{2} x^{2} + 3 \, a^{5} b \log \left (x^{2}\right ) - \frac {a^{6}}{2 \, x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=\frac {1}{10} \, b^{6} x^{10} + \frac {3}{4} \, a b^{5} x^{8} + \frac {5}{2} \, a^{2} b^{4} x^{6} + 5 \, a^{3} b^{3} x^{4} + \frac {15}{2} \, a^{4} b^{2} x^{2} + 3 \, a^{5} b \log \left (x^{2}\right ) - \frac {6 \, a^{5} b x^{2} + a^{6}}{2 \, x^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^3} \, dx=\frac {b^6\,x^{10}}{10}-\frac {a^6}{2\,x^2}+\frac {3\,a\,b^5\,x^8}{4}+6\,a^5\,b\,\ln \left (x\right )+\frac {15\,a^4\,b^2\,x^2}{2}+5\,a^3\,b^3\,x^4+\frac {5\,a^2\,b^4\,x^6}{2} \]
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