Optimal. Leaf size=128 \[ -\frac {a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}+\frac {a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}-\frac {3 a x^2 (A b-2 a B)}{2 b^5}+\frac {x^4 (A b-3 a B)}{4 b^4}+\frac {B x^6}{6 b^3} \]
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Rubi [A] time = 0.17, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} \frac {a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}-\frac {a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}+\frac {x^4 (A b-3 a B)}{4 b^4}-\frac {3 a x^2 (A b-2 a B)}{2 b^5}+\frac {B x^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {3 a (-A b+2 a B)}{b^5}+\frac {(A b-3 a B) x}{b^4}+\frac {B x^2}{b^3}-\frac {a^4 (-A b+a B)}{b^5 (a+b x)^3}+\frac {a^3 (-4 A b+5 a B)}{b^5 (a+b x)^2}-\frac {2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a (A b-2 a B) x^2}{2 b^5}+\frac {(A b-3 a B) x^4}{4 b^4}+\frac {B x^6}{6 b^3}-\frac {a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}+\frac {a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 116, normalized size = 0.91 \begin {gather*} \frac {\frac {3 a^4 (a B-A b)}{\left (a+b x^2\right )^2}+\frac {6 a^3 (4 A b-5 a B)}{a+b x^2}+12 a^2 (3 A b-5 a B) \log \left (a+b x^2\right )+3 b^2 x^4 (A b-3 a B)+18 a b x^2 (2 a B-A b)+2 b^3 B x^6}{12 b^6} \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^9 \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 = 0.43, size = 205, normalized size = 1.60 \begin {gather*} \frac {2 \, B b^{5} x^{10} - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{8} + 4 \, {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{6} - 27 \, B a^{5} + 21 \, A a^{4} b + 3 \, {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 6 \, {\left (B a^{4} b + A a^{3} b^{2}\right )} x^{2} - 12 \, {\left (5 \, B a^{5} - 3 \, A a^{4} b + {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{4} + 2 \, {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 159, normalized size = 1.24 \begin {gather*} -\frac {{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac {30 \, B a^{3} b^{2} x^{4} - 18 \, A a^{2} b^{3} x^{4} + 50 \, B a^{4} b x^{2} - 28 \, A a^{3} b^{2} x^{2} + 21 \, B a^{5} - 11 \, A a^{4} b}{4 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {2 \, B b^{6} x^{6} - 9 \, B a b^{5} x^{4} + 3 \, A b^{6} x^{4} + 36 \, B a^{2} b^{4} x^{2} - 18 \, A a b^{5} x^{2}}{12 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 158, normalized size = 1.23 \begin {gather*} \frac {B \,x^{6}}{6 b^{3}}+\frac {A \,x^{4}}{4 b^{3}}-\frac {3 B a \,x^{4}}{4 b^{4}}-\frac {A \,a^{4}}{4 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {3 A a \,x^{2}}{2 b^{4}}+\frac {B \,a^{5}}{4 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {3 B \,a^{2} x^{2}}{b^{5}}+\frac {2 A \,a^{3}}{\left (b \,x^{2}+a \right ) b^{5}}+\frac {3 A \,a^{2} \ln \left (b \,x^{2}+a \right )}{b^{5}}-\frac {5 B \,a^{4}}{2 \left (b \,x^{2}+a \right ) b^{6}}-\frac {5 B \,a^{3} \ln \left (b \,x^{2}+a \right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 141, normalized size = 1.10 \begin {gather*} -\frac {9 \, B a^{5} - 7 \, A a^{4} b + 2 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}}{4 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {2 \, B b^{2} x^{6} - 3 \, {\left (3 \, B a b - A b^{2}\right )} x^{4} + 18 \, {\left (2 \, B a^{2} - A a b\right )} x^{2}}{12 \, b^{5}} - \frac {{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x^{2} + a\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 155, normalized size = 1.21 \begin {gather*} x^4\,\left (\frac {A}{4\,b^3}-\frac {3\,B\,a}{4\,b^4}\right )-\frac {\frac {9\,B\,a^5-7\,A\,a^4\,b}{4\,b}+x^2\,\left (\frac {5\,B\,a^4}{2}-2\,A\,a^3\,b\right )}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}-x^2\,\left (\frac {3\,a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{2\,b}+\frac {3\,B\,a^2}{2\,b^5}\right )+\frac {B\,x^6}{6\,b^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (5\,B\,a^3-3\,A\,a^2\,b\right )}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.51, size = 143, normalized size = 1.12 \begin {gather*} \frac {B x^{6}}{6 b^{3}} - \frac {a^{2} \left (- 3 A b + 5 B a\right ) \log {\left (a + b x^{2} \right )}}{b^{6}} + x^{4} \left (\frac {A}{4 b^{3}} - \frac {3 B a}{4 b^{4}}\right ) + x^{2} \left (- \frac {3 A a}{2 b^{4}} + \frac {3 B a^{2}}{b^{5}}\right ) + \frac {7 A a^{4} b - 9 B a^{5} + x^{2} \left (8 A a^{3} b^{2} - 10 B a^{4} b\right )}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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