Optimal. Leaf size=158 \[ -\frac {9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}-\frac {a^4 x (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}+\frac {2 a^2 x (3 A b-5 a B)}{b^6}-\frac {a x^3 (A b-2 a B)}{b^5}+\frac {x^5 (A b-3 a B)}{5 b^4}+\frac {B x^7}{7 b^3} \]
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Rubi [A] time = 0.28, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {455, 1814, 1810, 205} \begin {gather*} \frac {a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}-\frac {a^4 x (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {2 a^2 x (3 A b-5 a B)}{b^6}-\frac {9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}+\frac {x^5 (A b-3 a B)}{5 b^4}-\frac {a x^3 (A b-2 a B)}{b^5}+\frac {B x^7}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 455
Rule 1810
Rule 1814
Rubi steps
\begin {align*} \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}-\frac {\int \frac {-a^4 (A b-a B)+4 a^3 b (A b-a B) x^2-4 a^2 b^2 (A b-a B) x^4+4 a b^3 (A b-a B) x^6-4 b^4 (A b-a B) x^8-4 b^5 B x^{10}}{\left (a+b x^2\right )^2} \, dx}{4 b^6}\\ &=-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \frac {-a^4 (15 A b-19 a B)+8 a^3 b (3 A b-4 a B) x^2-8 a^2 b^2 (2 A b-3 a B) x^4+8 a b^3 (A b-2 a B) x^6+8 a b^4 B x^8}{a+b x^2} \, dx}{8 a b^6}\\ &=-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \left (16 a^3 (3 A b-5 a B)-24 a^2 b (A b-2 a B) x^2+8 a b^2 (A b-3 a B) x^4+8 a b^3 B x^6+\frac {9 \left (-7 a^4 A b+11 a^5 B\right )}{a+b x^2}\right ) \, dx}{8 a b^6}\\ &=\frac {2 a^2 (3 A b-5 a B) x}{b^6}-\frac {a (A b-2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac {\left (9 a^3 (7 A b-11 a B)\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^6}\\ &=\frac {2 a^2 (3 A b-5 a B) x}{b^6}-\frac {a (A b-2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac {9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 158, normalized size = 1.00 \begin {gather*} \frac {9 a^{5/2} (11 a B-7 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}+\frac {a^4 x (a B-A b)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}-\frac {2 a^2 x (5 a B-3 A b)}{b^6}+\frac {a x^3 (2 a B-A b)}{b^5}+\frac {x^5 (A b-3 a B)}{5 b^4}+\frac {B x^7}{7 b^3} \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^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.47, size = 468, normalized size = 2.96 \begin {gather*} \left [\frac {80 \, B b^{5} x^{11} - 16 \, {\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 48 \, {\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 336 \, {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 1050 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} - 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b + {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{560 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {40 \, B b^{5} x^{11} - 8 \, {\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 24 \, {\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 168 \, {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 525 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} + 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b + {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{280 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 162, normalized size = 1.03 \begin {gather*} \frac {9 \, {\left (11 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} - \frac {21 \, B a^{4} b x^{3} - 17 \, A a^{3} b^{2} x^{3} + 19 \, B a^{5} x - 15 \, A a^{4} b x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {5 \, B b^{18} x^{7} - 21 \, B a b^{17} x^{5} + 7 \, A b^{18} x^{5} + 70 \, B a^{2} b^{16} x^{3} - 35 \, A a b^{17} x^{3} - 350 \, B a^{3} b^{15} x + 210 \, A a^{2} b^{16} x}{35 \, b^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 198, normalized size = 1.25 \begin {gather*} \frac {B \,x^{7}}{7 b^{3}}+\frac {17 A \,a^{3} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {A \,x^{5}}{5 b^{3}}-\frac {21 B \,a^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {3 B a \,x^{5}}{5 b^{4}}+\frac {15 A \,a^{4} x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {A a \,x^{3}}{b^{4}}-\frac {19 B \,a^{5} x}{8 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {2 B \,a^{2} x^{3}}{b^{5}}-\frac {63 A \,a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {99 B \,a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{6}}+\frac {6 A \,a^{2} x}{b^{5}}-\frac {10 B \,a^{3} x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.35, size = 171, normalized size = 1.08 \begin {gather*} -\frac {{\left (21 \, B a^{4} b - 17 \, A a^{3} b^{2}\right )} x^{3} + {\left (19 \, B a^{5} - 15 \, A a^{4} b\right )} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {9 \, {\left (11 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {5 \, B b^{3} x^{7} - 7 \, {\left (3 \, B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{3} - 70 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} x}{35 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 246, normalized size = 1.56 \begin {gather*} x^5\,\left (\frac {A}{5\,b^3}-\frac {3\,B\,a}{5\,b^4}\right )-\frac {x\,\left (\frac {19\,B\,a^5}{8}-\frac {15\,A\,a^4\,b}{8}\right )-x^3\,\left (\frac {17\,A\,a^3\,b^2}{8}-\frac {21\,B\,a^4\,b}{8}\right )}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}-x^3\,\left (\frac {a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b}+\frac {B\,a^2}{b^5}\right )-x\,\left (\frac {B\,a^3}{b^6}-\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b}+\frac {3\,B\,a^2}{b^5}\right )}{b}+\frac {3\,a^2\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b^2}\right )+\frac {B\,x^7}{7\,b^3}+\frac {9\,a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (7\,A\,b-11\,B\,a\right )}{11\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-11\,B\,a\right )}{8\,b^{13/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.44, size = 280, normalized size = 1.77 \begin {gather*} \frac {B x^{7}}{7 b^{3}} + x^{5} \left (\frac {A}{5 b^{3}} - \frac {3 B a}{5 b^{4}}\right ) + x^{3} \left (- \frac {A a}{b^{4}} + \frac {2 B a^{2}}{b^{5}}\right ) + x \left (\frac {6 A a^{2}}{b^{5}} - \frac {10 B a^{3}}{b^{6}}\right ) - \frac {9 \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log {\left (- \frac {9 b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} + \frac {9 \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log {\left (\frac {9 b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} + \frac {x^{3} \left (17 A a^{3} b^{2} - 21 B a^{4} b\right ) + x \left (15 A a^{4} b - 19 B a^{5}\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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