Optimal. Leaf size=67 \[ -\frac {1}{2} \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log \left (a+b x^2\right )+\frac {c^2 \log (x)}{a^2}+\frac {(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 67, 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} \[ -\frac {1}{2} \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log \left (a+b x^2\right )+\frac {c^2 \log (x)}{a^2}+\frac {(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^2}{x (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^2}{a^2 x}-\frac {(-b c+a d)^2}{a b (a+b x)^2}+\frac {-b^2 c^2+a^2 d^2}{a^2 b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {c^2 \log (x)}{a^2}-\frac {1}{2} \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log \left (a+b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 1.04 \[ \frac {\frac {(a d-b c) \left (\left (a+b x^2\right ) (a d+b c) \log \left (a+b x^2\right )+a (a d-b c)\right )}{b^2 \left (a+b x^2\right )}+2 c^2 \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 117, normalized size = 1.75 \[ \frac {a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} - {\left (a b^{2} c^{2} - a^{3} d^{2} + {\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (b^{3} c^{2} x^{2} + a b^{2} c^{2}\right )} \log \relax (x)}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 99, normalized size = 1.48 \[ \frac {c^{2} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{2}} + \frac {b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \, {\left (b x^{2} + a\right )} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 1.40 \[ \frac {a \,d^{2}}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {c^{2}}{2 \left (b \,x^{2}+a \right ) a}+\frac {c^{2} \ln \relax (x )}{a^{2}}-\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {c d}{\left (b \,x^{2}+a \right ) b}+\frac {d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 86, normalized size = 1.28 \[ \frac {c^{2} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} - \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 80, normalized size = 1.19 \[ \frac {c^2\,\ln \relax (x)}{a^2}+\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{2\,a\,b^2\,\left (b\,x^2+a\right )}+\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-b^2\,c^2\right )}{2\,a^2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 80, normalized size = 1.19 \[ \frac {a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac {c^{2} \log {\relax (x )}}{a^{2}} + \frac {\left (a d - b c\right ) \left (a d + b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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