Optimal. Leaf size=62 \[ -\frac {(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac {b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac {b^2 x^2}{2 d^2} \]
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Rubi [A] time = 0.06, antiderivative size = 62, 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 = {444, 43} \begin {gather*} -\frac {(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac {b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac {b^2 x^2}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{(c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{d^2}+\frac {(-b c+a d)^2}{d^2 (c+d x)^2}-\frac {2 b (b c-a d)}{d^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^2}{2 d^2}-\frac {(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac {b (b c-a d) \log \left (c+d x^2\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.90 \begin {gather*} \frac {-\frac {(b c-a d)^2}{c+d x^2}+2 b (a d-b c) \log \left (c+d x^2\right )+b^2 d x^2}{2 d^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 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.80, size = 101, normalized size = 1.63 \begin {gather*} \frac {b^{2} d^{2} x^{4} + b^{2} c d x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 110, normalized size = 1.77 \begin {gather*} \frac {{\left (d x^{2} + c\right )} b^{2}}{2 \, d^{3}} + \frac {{\left (b^{2} c - a b d\right )} \log \left (\frac {{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2} {\left | d \right |}}\right )}{d^{3}} - \frac {\frac {b^{2} c^{2} d}{d x^{2} + c} - \frac {2 \, a b c d^{2}}{d x^{2} + c} + \frac {a^{2} d^{3}}{d x^{2} + c}}{2 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 1.56 \begin {gather*} \frac {b^{2} x^{2}}{2 d^{2}}-\frac {a^{2}}{2 \left (d \,x^{2}+c \right ) d}+\frac {a b c}{\left (d \,x^{2}+c \right ) d^{2}}+\frac {a b \ln \left (d \,x^{2}+c \right )}{d^{2}}-\frac {b^{2} c^{2}}{2 \left (d \,x^{2}+c \right ) d^{3}}-\frac {b^{2} c \ln \left (d \,x^{2}+c \right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 74, normalized size = 1.19 \begin {gather*} \frac {b^{2} x^{2}}{2 \, d^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} - \frac {{\left (b^{2} c - a b d\right )} \log \left (d x^{2} + c\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 77, normalized size = 1.24 \begin {gather*} \frac {b^2\,x^2}{2\,d^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{2\,d\,\left (d^3\,x^2+c\,d^2\right )}-\frac {\ln \left (d\,x^2+c\right )\,\left (b^2\,c-a\,b\,d\right )}{d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 68, normalized size = 1.10 \begin {gather*} \frac {b^{2} x^{2}}{2 d^{2}} + \frac {b \left (a d - b c\right ) \log {\left (c + d x^{2} \right )}}{d^{3}} + \frac {- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}}{2 c d^{3} + 2 d^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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