Optimal. Leaf size=106 \[ -\frac {a^2}{2 c^3 x^2}-\frac {a (2 b c-3 a d) \log \left (c+d x^2\right )}{2 c^4}+\frac {a \log (x) (2 b c-3 a d)}{c^4}+\frac {a (b c-a d)}{c^3 \left (c+d x^2\right )}-\frac {(b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 106, 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} \[ -\frac {a^2}{2 c^3 x^2}+\frac {a (b c-a d)}{c^3 \left (c+d x^2\right )}-\frac {(b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2}-\frac {a (2 b c-3 a d) \log \left (c+d x^2\right )}{2 c^4}+\frac {a \log (x) (2 b c-3 a d)}{c^4} \]
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 )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{c^3 x^2}-\frac {a (-2 b c+3 a d)}{c^4 x}+\frac {(b c-a d)^2}{c^2 (c+d x)^3}+\frac {2 a d (-b c+a d)}{c^3 (c+d x)^2}+\frac {a d (-2 b c+3 a d)}{c^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{2 c^3 x^2}-\frac {(b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2}+\frac {a (b c-a d)}{c^3 \left (c+d x^2\right )}+\frac {a (2 b c-3 a d) \log (x)}{c^4}-\frac {a (2 b c-3 a d) \log \left (c+d x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 99, normalized size = 0.93 \[ \frac {-\frac {2 a^2 c}{x^2}-\frac {c^2 (b c-a d)^2}{d \left (c+d x^2\right )^2}+\frac {4 a c (b c-a d)}{c+d x^2}+2 a (3 a d-2 b c) \log \left (c+d x^2\right )+4 a \log (x) (2 b c-3 a d)}{4 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 256, normalized size = 2.42 \[ -\frac {2 \, a^{2} c^{3} d - 2 \, {\left (2 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x^{4} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 9 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \, {\left ({\left (2 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{6} + 2 \, {\left (2 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x^{4} + {\left (2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (2 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{6} + 2 \, {\left (2 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x^{4} + {\left (2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \relax (x)}{4 \, {\left (c^{4} d^{3} x^{6} + 2 \, c^{5} d^{2} x^{4} + c^{6} d x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 177, normalized size = 1.67 \[ \frac {{\left (2 \, a b c - 3 \, a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{4}} - \frac {{\left (2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{4} d} - \frac {2 \, a b c x^{2} - 3 \, a^{2} d x^{2} + a^{2} c}{2 \, c^{4} x^{2}} + \frac {6 \, a b c d^{3} x^{4} - 9 \, a^{2} d^{4} x^{4} + 16 \, a b c^{2} d^{2} x^{2} - 22 \, a^{2} c d^{3} x^{2} - b^{2} c^{4} + 12 \, a b c^{3} d - 14 \, a^{2} c^{2} d^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} c^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 149, normalized size = 1.41 \[ -\frac {a^{2} d}{4 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {a b}{2 \left (d \,x^{2}+c \right )^{2} c}-\frac {b^{2}}{4 \left (d \,x^{2}+c \right )^{2} d}-\frac {a^{2} d}{\left (d \,x^{2}+c \right ) c^{3}}-\frac {3 a^{2} d \ln \relax (x )}{c^{4}}+\frac {3 a^{2} d \ln \left (d \,x^{2}+c \right )}{2 c^{4}}+\frac {a b}{\left (d \,x^{2}+c \right ) c^{2}}+\frac {2 a b \ln \relax (x )}{c^{3}}-\frac {a b \ln \left (d \,x^{2}+c \right )}{c^{3}}-\frac {a^{2}}{2 c^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 142, normalized size = 1.34 \[ -\frac {2 \, a^{2} c^{2} d - 2 \, {\left (2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 9 \, a^{2} c d^{2}\right )} x^{2}}{4 \, {\left (c^{3} d^{3} x^{6} + 2 \, c^{4} d^{2} x^{4} + c^{5} d x^{2}\right )}} - \frac {{\left (2 \, a b c - 3 \, a^{2} d\right )} \log \left (d x^{2} + c\right )}{2 \, c^{4}} + \frac {{\left (2 \, a b c - 3 \, a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 132, normalized size = 1.25 \[ \frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d-2\,a\,b\,c\right )}{2\,c^4}-\frac {\frac {a^2}{2\,c}+\frac {x^2\,\left (9\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{4\,c^2\,d}+\frac {a\,d\,x^4\,\left (3\,a\,d-2\,b\,c\right )}{2\,c^3}}{c^2\,x^2+2\,c\,d\,x^4+d^2\,x^6}-\frac {\ln \relax (x)\,\left (3\,a^2\,d-2\,a\,b\,c\right )}{c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 139, normalized size = 1.31 \[ - \frac {a \left (3 a d - 2 b c\right ) \log {\relax (x )}}{c^{4}} + \frac {a \left (3 a d - 2 b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{4}} + \frac {- 2 a^{2} c^{2} d + x^{4} \left (- 6 a^{2} d^{3} + 4 a b c d^{2}\right ) + x^{2} \left (- 9 a^{2} c d^{2} + 6 a b c^{2} d - b^{2} c^{3}\right )}{4 c^{5} d x^{2} + 8 c^{4} d^{2} x^{4} + 4 c^{3} d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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