Optimal. Leaf size=58 \[ -\frac {a^2}{2 c x^2}+\frac {a (2 b c-a d) \log (x)}{c^2}+\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d} \]
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Rubi [A]
time = 0.04, 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 = {457, 90}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
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} \text {Subst}\left (\int \frac {(a+b x)^2}{x^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.02, size = 60, normalized size = 1.03 \begin {gather*} \frac {-a^2 c d-2 a d (-2 b c+a d) x^2 \log (x)+(b c-a d)^2 x^2 \log \left (c+d x^2\right )}{2 c^2 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 66, normalized size = 1.14
method | result | size |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{2} d}-\frac {a^{2}}{2 c \,x^{2}}-\frac {a \left (a d -2 b c \right ) \ln \left (x \right )}{c^{2}}\) | \(66\) |
norman | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{2} d}-\frac {a^{2}}{2 c \,x^{2}}-\frac {a \left (a d -2 b c \right ) \ln \left (x \right )}{c^{2}}\) | \(66\) |
risch | \(-\frac {a^{2}}{2 c \,x^{2}}-\frac {a^{2} \ln \left (x \right ) d}{c^{2}}+\frac {2 a \ln \left (x \right ) b}{c}+\frac {d \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{2}}-\frac {\ln \left (-d \,x^{2}-c \right ) a b}{c}+\frac {\ln \left (-d \,x^{2}-c \right ) b^{2}}{2 d}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 70, normalized size = 1.21 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 74, normalized size = 1.28 \begin {gather*} -\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 \left (x\right )}{2 \, c^{2} d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.84, size = 49, normalized size = 0.84 \begin {gather*} - \frac {a^{2}}{2 c x^{2}} - \frac {a \left (a d - 2 b c\right ) \log {\left (x \right )}}{c^{2}} + \frac {\left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 91, normalized size = 1.57 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 67, normalized size = 1.16 \begin {gather*} \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 \left (x\right )\,\left (a^2\,d-2\,a\,b\,c\right )}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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