Optimal. Leaf size=51 \[ \frac {a^2 \log (x)}{c}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac {b^2 x^2}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 72} \[ \frac {a^2 \log (x)}{c}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac {b^2 x^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{d}+\frac {a^2}{c x}-\frac {(b c-a d)^2}{c d (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^2}{2 d}+\frac {a^2 \log (x)}{c}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.98 \[ \frac {2 a^2 d^2 \log (x)-(b c-a d)^2 \log \left (c+d x^2\right )+b^2 c d x^2}{2 c d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 59, normalized size = 1.16 \[ \frac {b^{2} c d x^{2} + 2 \, a^{2} d^{2} \log \relax (x) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 62, normalized size = 1.22 \[ \frac {b^{2} x^{2}}{2 \, d} + \frac {a^{2} \log \left (x^{2}\right )}{2 \, c} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 1.35 \[ \frac {b^{2} x^{2}}{2 d}+\frac {a^{2} \ln \relax (x )}{c}-\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 c}+\frac {a b \ln \left (d \,x^{2}+c \right )}{d}-\frac {b^{2} c \ln \left (d \,x^{2}+c \right )}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 61, normalized size = 1.20 \[ \frac {b^{2} x^{2}}{2 \, d} + \frac {a^{2} \log \left (x^{2}\right )}{2 \, c} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 58, normalized size = 1.14 \[ \frac {b^2\,x^2}{2\,d}+\frac {a^2\,\ln \relax (x)}{c}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 41, normalized size = 0.80 \[ \frac {a^{2} \log {\relax (x )}}{c} + \frac {b^{2} x^{2}}{2 d} - \frac {\left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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