Optimal. Leaf size=61 \[ \frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}-\frac {b x^2 (b c-a d)}{2 d^2}+\frac {\left (a+b x^2\right )^2}{4 d} \]
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Rubi [A] time = 0.05, antiderivative size = 61, 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 x^2 (b c-a d)}{2 d^2}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}+\frac {\left (a+b x^2\right )^2}{4 d} \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}{c+d x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{c+d x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b (b c-a d) x^2}{2 d^2}+\frac {\left (a+b x^2\right )^2}{4 d}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.80 \begin {gather*} \frac {b d x^2 \left (4 a d-2 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (c+d x^2\right )}{4 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}{c+d x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.90, size = 66, normalized size = 1.08 \begin {gather*} \frac {b^{2} d^{2} x^{4} - 2 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 67, normalized size = 1.10 \begin {gather*} \frac {b^{2} d x^{4} - 2 \, b^{2} c x^{2} + 4 \, a b d x^{2}}{4 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 85, normalized size = 1.39 \begin {gather*} \frac {b^{2} x^{4}}{4 d}+\frac {a b \,x^{2}}{d}-\frac {b^{2} c \,x^{2}}{2 d^{2}}+\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 d}-\frac {a b c \ln \left (d \,x^{2}+c \right )}{d^{2}}+\frac {b^{2} c^{2} \ln \left (d \,x^{2}+c \right )}{2 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 65, normalized size = 1.07 \begin {gather*} \frac {b^{2} d x^{4} - 2 \, {\left (b^{2} c - 2 \, a b d\right )} x^{2}}{4 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 68, normalized size = 1.11 \begin {gather*} \frac {b^2\,x^4}{4\,d}-x^2\,\left (\frac {b^2\,c}{2\,d^2}-\frac {a\,b}{d}\right )+\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 49, normalized size = 0.80 \begin {gather*} \frac {b^{2} x^{4}}{4 d} + x^{2} \left (\frac {a b}{d} - \frac {b^{2} c}{2 d^{2}}\right ) + \frac {\left (a d - b c\right )^{2} \log {\left (c + d x^{2} \right )}}{2 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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