Optimal. Leaf size=106 \[ \frac {105 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{11/2}}+\frac {105 b}{16 a^5 x}-\frac {35}{16 a^4 x^3}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.07, antiderivative size = 106, 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, 290, 325, 205} \[ \frac {105 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{11/2}}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {105 b}{16 a^5 x}-\frac {35}{16 a^4 x^3}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {\left (3 b^3\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^3} \, dx}{2 a}\\ &=\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {\left (21 b^2\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}+\frac {(105 b) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{16 a^3}\\ &=-\frac {35}{16 a^4 x^3}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}-\frac {\left (105 b^2\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{16 a^4}\\ &=-\frac {35}{16 a^4 x^3}+\frac {105 b}{16 a^5 x}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}+\frac {\left (105 b^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{16 a^5}\\ &=-\frac {35}{16 a^4 x^3}+\frac {105 b}{16 a^5 x}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}+\frac {3}{8 a^2 x^3 \left (a+b x^2\right )^2}+\frac {21}{16 a^3 x^3 \left (a+b x^2\right )}+\frac {105 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.86 \[ \frac {\frac {\sqrt {a} \left (-16 a^4+144 a^3 b x^2+693 a^2 b^2 x^4+840 a b^3 x^6+315 b^4 x^8\right )}{x^3 \left (a+b x^2\right )^3}+315 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48 a^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 304, normalized size = 2.87 \[ \left [\frac {630 \, b^{4} x^{8} + 1680 \, a b^{3} x^{6} + 1386 \, a^{2} b^{2} x^{4} + 288 \, a^{3} b x^{2} - 32 \, a^{4} + 315 \, {\left (b^{4} x^{9} + 3 \, a b^{3} x^{7} + 3 \, a^{2} b^{2} x^{5} + a^{3} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{96 \, {\left (a^{5} b^{3} x^{9} + 3 \, a^{6} b^{2} x^{7} + 3 \, a^{7} b x^{5} + a^{8} x^{3}\right )}}, \frac {315 \, b^{4} x^{8} + 840 \, a b^{3} x^{6} + 693 \, a^{2} b^{2} x^{4} + 144 \, a^{3} b x^{2} - 16 \, a^{4} + 315 \, {\left (b^{4} x^{9} + 3 \, a b^{3} x^{7} + 3 \, a^{2} b^{2} x^{5} + a^{3} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{48 \, {\left (a^{5} b^{3} x^{9} + 3 \, a^{6} b^{2} x^{7} + 3 \, a^{7} b x^{5} + a^{8} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 82, normalized size = 0.77 \[ \frac {105 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{5}} + \frac {315 \, b^{4} x^{8} + 840 \, a b^{3} x^{6} + 693 \, a^{2} b^{2} x^{4} + 144 \, a^{3} b x^{2} - 16 \, a^{4}}{48 \, {\left (b x^{3} + a x\right )}^{3} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 0.93 \[ \frac {41 b^{4} x^{5}}{16 \left (b \,x^{2}+a \right )^{3} a^{5}}+\frac {35 b^{3} x^{3}}{6 \left (b \,x^{2}+a \right )^{3} a^{4}}+\frac {55 b^{2} x}{16 \left (b \,x^{2}+a \right )^{3} a^{3}}+\frac {105 b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \sqrt {a b}\, a^{5}}+\frac {4 b}{a^{5} x}-\frac {1}{3 a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 108, normalized size = 1.02 \[ \frac {315 \, b^{4} x^{8} + 840 \, a b^{3} x^{6} + 693 \, a^{2} b^{2} x^{4} + 144 \, a^{3} b x^{2} - 16 \, a^{4}}{48 \, {\left (a^{5} b^{3} x^{9} + 3 \, a^{6} b^{2} x^{7} + 3 \, a^{7} b x^{5} + a^{8} x^{3}\right )}} + \frac {105 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 102, normalized size = 0.96 \[ \frac {\frac {3\,b\,x^2}{a^2}-\frac {1}{3\,a}+\frac {231\,b^2\,x^4}{16\,a^3}+\frac {35\,b^3\,x^6}{2\,a^4}+\frac {105\,b^4\,x^8}{16\,a^5}}{a^3\,x^3+3\,a^2\,b\,x^5+3\,a\,b^2\,x^7+b^3\,x^9}+\frac {105\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,a^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 162, normalized size = 1.53 \[ - \frac {105 \sqrt {- \frac {b^{3}}{a^{11}}} \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}}}{b^{2}} + x \right )}}{32} + \frac {105 \sqrt {- \frac {b^{3}}{a^{11}}} \log {\left (\frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}}}{b^{2}} + x \right )}}{32} + \frac {- 16 a^{4} + 144 a^{3} b x^{2} + 693 a^{2} b^{2} x^{4} + 840 a b^{3} x^{6} + 315 b^{4} x^{8}}{48 a^{8} x^{3} + 144 a^{7} b x^{5} + 144 a^{6} b^{2} x^{7} + 48 a^{5} b^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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