Optimal. Leaf size=122 \[ -\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac {7 x}{256 a b^4 \left (a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 294, 205,
211} \begin {gather*} \frac {7 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}}+\frac {7 x}{256 a b^4 \left (a+b x^2\right )}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {x^7}{10 b \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 211
Rule 294
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^8}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} \left (7 b^4\right ) \int \frac {x^6}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}+\frac {1}{16} \left (7 b^2\right ) \int \frac {x^4}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}+\frac {7}{32} \int \frac {x^2}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac {7 \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac {7 x}{256 a b^4 \left (a+b x^2\right )}+\frac {7 \int \frac {1}{a b+b^2 x^2} \, dx}{256 a b^3}\\ &=-\frac {x^7}{10 b \left (a+b x^2\right )^5}-\frac {7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac {7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac {7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac {7 x}{256 a b^4 \left (a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 91, normalized size = 0.75 \begin {gather*} -\frac {x \left (105 a^4+490 a^3 b x^2+896 a^2 b^2 x^4+790 a b^3 x^6-105 b^4 x^8\right )}{3840 a b^4 \left (a+b x^2\right )^5}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{3/2} b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 80, normalized size = 0.66
method | result | size |
default | \(\frac {\frac {7 x^{9}}{256 a}-\frac {79 x^{7}}{384 b}-\frac {7 a \,x^{5}}{30 b^{2}}-\frac {49 a^{2} x^{3}}{384 b^{3}}-\frac {7 a^{3} x}{256 b^{4}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 b^{4} a \sqrt {a b}}\) | \(80\) |
risch | \(\frac {\frac {7 x^{9}}{256 a}-\frac {79 x^{7}}{384 b}-\frac {7 a \,x^{5}}{30 b^{2}}-\frac {49 a^{2} x^{3}}{384 b^{3}}-\frac {7 a^{3} x}{256 b^{4}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}-\frac {7 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{4} a}+\frac {7 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{4} a}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 131, normalized size = 1.07 \begin {gather*} \frac {105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} + \frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 390, normalized size = 3.20 \begin {gather*} \left [\frac {210 \, a b^{5} x^{9} - 1580 \, a^{2} b^{4} x^{7} - 1792 \, a^{3} b^{3} x^{5} - 980 \, a^{4} b^{2} x^{3} - 210 \, a^{5} b x - 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, \frac {105 \, a b^{5} x^{9} - 790 \, a^{2} b^{4} x^{7} - 896 \, a^{3} b^{3} x^{5} - 490 \, a^{4} b^{2} x^{3} - 105 \, a^{5} b x + 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{3840 \, {\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 194, normalized size = 1.59 \begin {gather*} - \frac {7 \sqrt {- \frac {1}{a^{3} b^{9}}} \log {\left (- a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} + x \right )}}{512} + \frac {7 \sqrt {- \frac {1}{a^{3} b^{9}}} \log {\left (a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} + x \right )}}{512} + \frac {- 105 a^{4} x - 490 a^{3} b x^{3} - 896 a^{2} b^{2} x^{5} - 790 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{6} b^{4} + 19200 a^{5} b^{5} x^{2} + 38400 a^{4} b^{6} x^{4} + 38400 a^{3} b^{7} x^{6} + 19200 a^{2} b^{8} x^{8} + 3840 a b^{9} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.48, size = 84, normalized size = 0.69 \begin {gather*} \frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a b^{4}} + \frac {105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.42, size = 119, normalized size = 0.98 \begin {gather*} \frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{3/2}\,b^{9/2}}-\frac {\frac {79\,x^7}{384\,b}-\frac {7\,x^9}{256\,a}+\frac {7\,a\,x^5}{30\,b^2}+\frac {7\,a^3\,x}{256\,b^4}+\frac {49\,a^2\,x^3}{384\,b^3}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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