Optimal. Leaf size=58 \[ -\frac {a^2}{2 c x^2}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}+\frac {a \log (x) (2 b c-a d)}{c^2} \]
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Rubi [A] time = 0.06, antiderivative size = 58, 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, 88} \[ -\frac {a^2}{2 c x^2}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}+\frac {a \log (x) (2 b c-a d)}{c^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{c x^2}-\frac {a (-2 b c+a d)}{c^2 x}+\frac {(b c-a d)^2}{c^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{2 c x^2}+\frac {a (2 b c-a d) \log (x)}{c^2}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 1.03 \[ \frac {a^2 (-c) d-2 a d x^2 \log (x) (a d-2 b c)+x^2 (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 74, normalized size = 1.28 \[ -\frac {a^{2} c d - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (d x^{2} + c\right ) - 2 \, {\left (2 \, a b c d - a^{2} d^{2}\right )} x^{2} \log \relax (x)}{2 \, c^{2} d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 91, normalized size = 1.57 \[ \frac {{\left (2 \, a b c - a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d} - \frac {2 \, a b c x^{2} - a^{2} d x^{2} + a^{2} c}{2 \, c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 1.40 \[ -\frac {a^{2} d \ln \relax (x )}{c^{2}}+\frac {a^{2} d \ln \left (d \,x^{2}+c \right )}{2 c^{2}}+\frac {2 a b \ln \relax (x )}{c}-\frac {a b \ln \left (d \,x^{2}+c \right )}{c}+\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{2 d}-\frac {a^{2}}{2 c \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 70, normalized size = 1.21 \[ \frac {{\left (2 \, a b c - a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{2}} - \frac {a^{2}}{2 \, c x^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 67, normalized size = 1.16 \[ \frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}-\frac {a^2}{2\,c\,x^2}-\frac {\ln \relax (x)\,\left (a^2\,d-2\,a\,b\,c\right )}{c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 49, normalized size = 0.84 \[ - \frac {a^{2}}{2 c x^{2}} - \frac {a \left (a d - 2 b c\right ) \log {\relax (x )}}{c^{2}} + \frac {\left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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