Optimal. Leaf size=144 \[ \frac {3003 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{15/2}}+\frac {3003 b}{256 a^7 x}-\frac {1001}{256 a^6 x^3}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 290, 325, 205} \[ \frac {3003 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{15/2}}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {3003 b}{256 a^7 x}-\frac {1001}{256 a^6 x^3}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5} \]
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 )^3} \, dx &=b^6 \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {\left (13 b^5\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^5} \, dx}{10 a}\\ &=\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {\left (143 b^4\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^4} \, dx}{80 a^2}\\ &=\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {\left (429 b^3\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^3} \, dx}{160 a^3}\\ &=\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {\left (3003 b^2\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^2} \, dx}{640 a^4}\\ &=\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {(3003 b) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{256 a^5}\\ &=-\frac {1001}{256 a^6 x^3}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}-\frac {\left (3003 b^2\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{256 a^6}\\ &=-\frac {1001}{256 a^6 x^3}+\frac {3003 b}{256 a^7 x}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {\left (3003 b^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 a^7}\\ &=-\frac {1001}{256 a^6 x^3}+\frac {3003 b}{256 a^7 x}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}+\frac {13}{80 a^2 x^3 \left (a+b x^2\right )^4}+\frac {143}{480 a^3 x^3 \left (a+b x^2\right )^3}+\frac {429}{640 a^4 x^3 \left (a+b x^2\right )^2}+\frac {3003}{1280 a^5 x^3 \left (a+b x^2\right )}+\frac {3003 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{15/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 113, normalized size = 0.78 \[ \frac {\frac {\sqrt {a} \left (-1280 a^6+16640 a^5 b x^2+137995 a^4 b^2 x^4+338910 a^3 b^3 x^6+384384 a^2 b^4 x^8+210210 a b^5 x^{10}+45045 b^6 x^{12}\right )}{x^3 \left (a+b x^2\right )^5}+45045 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3840 a^{15/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 436, normalized size = 3.03 \[ \left [\frac {90090 \, b^{6} x^{12} + 420420 \, a b^{5} x^{10} + 768768 \, a^{2} b^{4} x^{8} + 677820 \, a^{3} b^{3} x^{6} + 275990 \, a^{4} b^{2} x^{4} + 33280 \, a^{5} b x^{2} - 2560 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}, \frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6} + 45045 \, {\left (b^{6} x^{13} + 5 \, a b^{5} x^{11} + 10 \, a^{2} b^{4} x^{9} + 10 \, a^{3} b^{3} x^{7} + 5 \, a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 104, normalized size = 0.72 \[ \frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} + \frac {18 \, b x^{2} - a}{3 \, a^{7} x^{3}} + \frac {22005 \, b^{6} x^{9} + 96290 \, a b^{5} x^{7} + 160384 \, a^{2} b^{4} x^{5} + 121310 \, a^{3} b^{3} x^{3} + 35595 \, a^{4} b^{2} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 139, normalized size = 0.97 \[ \frac {1467 b^{6} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} a^{7}}+\frac {9629 b^{5} x^{7}}{384 \left (b \,x^{2}+a \right )^{5} a^{6}}+\frac {1253 b^{4} x^{5}}{30 \left (b \,x^{2}+a \right )^{5} a^{5}}+\frac {12131 b^{3} x^{3}}{384 \left (b \,x^{2}+a \right )^{5} a^{4}}+\frac {2373 b^{2} x}{256 \left (b \,x^{2}+a \right )^{5} a^{3}}+\frac {3003 b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, a^{7}}+\frac {6 b}{a^{7} x}-\frac {1}{3 a^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.10, size = 152, normalized size = 1.06 \[ \frac {45045 \, b^{6} x^{12} + 210210 \, a b^{5} x^{10} + 384384 \, a^{2} b^{4} x^{8} + 338910 \, a^{3} b^{3} x^{6} + 137995 \, a^{4} b^{2} x^{4} + 16640 \, a^{5} b x^{2} - 1280 \, a^{6}}{3840 \, {\left (a^{7} b^{5} x^{13} + 5 \, a^{8} b^{4} x^{11} + 10 \, a^{9} b^{3} x^{9} + 10 \, a^{10} b^{2} x^{7} + 5 \, a^{11} b x^{5} + a^{12} x^{3}\right )}} + \frac {3003 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 146, normalized size = 1.01 \[ \frac {\frac {13\,b\,x^2}{3\,a^2}-\frac {1}{3\,a}+\frac {27599\,b^2\,x^4}{768\,a^3}+\frac {11297\,b^3\,x^6}{128\,a^4}+\frac {1001\,b^4\,x^8}{10\,a^5}+\frac {7007\,b^5\,x^{10}}{128\,a^6}+\frac {3003\,b^6\,x^{12}}{256\,a^7}}{a^5\,x^3+5\,a^4\,b\,x^5+10\,a^3\,b^2\,x^7+10\,a^2\,b^3\,x^9+5\,a\,b^4\,x^{11}+b^5\,x^{13}}+\frac {3003\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{15/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 209, normalized size = 1.45 \[ - \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (- \frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {3003 \sqrt {- \frac {b^{3}}{a^{15}}} \log {\left (\frac {a^{8} \sqrt {- \frac {b^{3}}{a^{15}}}}{b^{2}} + x \right )}}{512} + \frac {- 1280 a^{6} + 16640 a^{5} b x^{2} + 137995 a^{4} b^{2} x^{4} + 338910 a^{3} b^{3} x^{6} + 384384 a^{2} b^{4} x^{8} + 210210 a b^{5} x^{10} + 45045 b^{6} x^{12}}{3840 a^{12} x^{3} + 19200 a^{11} b x^{5} + 38400 a^{10} b^{2} x^{7} + 38400 a^{9} b^{3} x^{9} + 19200 a^{8} b^{4} x^{11} + 3840 a^{7} b^{5} x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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