Optimal. Leaf size=60 \[ \frac {a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {B x^2}{2 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 60, 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 (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {B x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {B}{b^2}+\frac {a (-A b+a B)}{b^2 (a+b x)^2}+\frac {A b-2 a B}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {B x^2}{2 b^2}+\frac {a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.83 \begin {gather*} \frac {\frac {a (A b-a B)}{a+b x^2}+(A b-2 a B) \log \left (a+b x^2\right )+b B x^2}{2 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^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 81, normalized size = 1.35 \begin {gather*} \frac {B b^{2} x^{4} + B a b x^{2} - B a^{2} + A a b - {\left (2 \, B a^{2} - A a b + {\left (2 \, B a b - A b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 91, normalized size = 1.52 \begin {gather*} \frac {\frac {{\left (b x^{2} + a\right )} B}{b^{2}} + \frac {{\left (2 \, B a - A b\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{2}} - \frac {\frac {B a^{2} b}{b x^{2} + a} - \frac {A a b^{2}}{b x^{2} + a}}{b^{3}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 1.23 \begin {gather*} \frac {B \,x^{2}}{2 b^{2}}+\frac {A a}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {A \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {B \,a^{2}}{2 \left (b \,x^{2}+a \right ) b^{3}}-\frac {B a \ln \left (b \,x^{2}+a \right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 60, normalized size = 1.00 \begin {gather*} \frac {B x^{2}}{2 \, b^{2}} - \frac {B a^{2} - A a b}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} - \frac {{\left (2 \, B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 62, normalized size = 1.03 \begin {gather*} \frac {B\,x^2}{2\,b^2}+\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b-2\,B\,a\right )}{2\,b^3}-\frac {B\,a^2-A\,a\,b}{2\,b\,\left (b^3\,x^2+a\,b^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 56, normalized size = 0.93 \begin {gather*} \frac {B x^{2}}{2 b^{2}} + \frac {A a b - B a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\left (- A b + 2 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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