Optimal. Leaf size=142 \[ \frac {3003 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}}-\frac {3003 a x}{256 b^7}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}+\frac {1001 x^3}{256 b^6} \]
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Rubi [A] time = 0.09, antiderivative size = 142, 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, 288, 302, 205} \begin {gather*} \frac {3003 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac {3003 a x}{256 b^7}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}+\frac {1001 x^3}{256 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^{14}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} \left (13 b^4\right ) \int \frac {x^{12}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}+\frac {1}{80} \left (143 b^2\right ) \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}+\frac {429}{160} \int \frac {x^8}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}+\frac {3003 \int \frac {x^6}{\left (a b+b^2 x^2\right )^2} \, dx}{640 b^2}\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 \int \frac {x^4}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 \int \left (-\frac {a}{b^3}+\frac {x^2}{b^2}+\frac {a^2}{b^2 \left (a b+b^2 x^2\right )}\right ) \, dx}{256 b^4}\\ &=-\frac {3003 a x}{256 b^7}+\frac {1001 x^3}{256 b^6}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {\left (3003 a^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 b^6}\\ &=-\frac {3003 a x}{256 b^7}+\frac {1001 x^3}{256 b^6}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 111, normalized size = 0.78 \begin {gather*} \frac {45045 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {\sqrt {b} x \left (-45045 a^6-210210 a^5 b x^2-384384 a^4 b^2 x^4-338910 a^3 b^3 x^6-137995 a^2 b^4 x^8-16640 a b^5 x^{10}+1280 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}}{3840 b^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.90, size = 428, normalized size = 3.01 \begin {gather*} \left [\frac {2560 \, b^{6} x^{13} - 33280 \, a b^{5} x^{11} - 275990 \, a^{2} b^{4} x^{9} - 677820 \, a^{3} b^{3} x^{7} - 768768 \, a^{4} b^{2} x^{5} - 420420 \, a^{5} b x^{3} - 90090 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{7680 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, \frac {1280 \, b^{6} x^{13} - 16640 \, a b^{5} x^{11} - 137995 \, a^{2} b^{4} x^{9} - 338910 \, a^{3} b^{3} x^{7} - 384384 \, a^{4} b^{2} x^{5} - 210210 \, a^{5} b x^{3} - 45045 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 106, normalized size = 0.75 \begin {gather*} \frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} - \frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b x^{2} + a\right )}^{5} b^{7}} + \frac {b^{12} x^{3} - 18 \, a b^{11} x}{3 \, b^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 137, normalized size = 0.96 \begin {gather*} -\frac {2373 a^{2} x^{9}}{256 \left (b \,x^{2}+a \right )^{5} b^{3}}-\frac {12131 a^{3} x^{7}}{384 \left (b \,x^{2}+a \right )^{5} b^{4}}-\frac {1253 a^{4} x^{5}}{30 \left (b \,x^{2}+a \right )^{5} b^{5}}-\frac {9629 a^{5} x^{3}}{384 \left (b \,x^{2}+a \right )^{5} b^{6}}-\frac {1467 a^{6} x}{256 \left (b \,x^{2}+a \right )^{5} b^{7}}+\frac {x^{3}}{3 b^{6}}+\frac {3003 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, b^{7}}-\frac {6 a x}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 148, normalized size = 1.04 \begin {gather*} -\frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} + \frac {b x^{3} - 18 \, a x}{3 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 143, normalized size = 1.01 \begin {gather*} \frac {x^3}{3\,b^6}-\frac {\frac {1467\,a^6\,x}{256}+\frac {9629\,a^5\,b\,x^3}{384}+\frac {1253\,a^4\,b^2\,x^5}{30}+\frac {12131\,a^3\,b^3\,x^7}{384}+\frac {2373\,a^2\,b^4\,x^9}{256}}{a^5\,b^7+5\,a^4\,b^8\,x^2+10\,a^3\,b^9\,x^4+10\,a^2\,b^{10}\,x^6+5\,a\,b^{11}\,x^8+b^{12}\,x^{10}}+\frac {3003\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,b^{15/2}}-\frac {6\,a\,x}{b^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 204, normalized size = 1.44 \begin {gather*} - \frac {6 a x}{b^{7}} - \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x - \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x + \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {- 22005 a^{6} x - 96290 a^{5} b x^{3} - 160384 a^{4} b^{2} x^{5} - 121310 a^{3} b^{3} x^{7} - 35595 a^{2} b^{4} x^{9}}{3840 a^{5} b^{7} + 19200 a^{4} b^{8} x^{2} + 38400 a^{3} b^{9} x^{4} + 38400 a^{2} b^{10} x^{6} + 19200 a b^{11} x^{8} + 3840 b^{12} x^{10}} + \frac {x^{3}}{3 b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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