Optimal. Leaf size=87 \[ -\frac {1}{2 a c x^2}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)}-\frac {d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)} \]
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Rubi [A]
time = 0.07, antiderivative size = 87, 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, 84}
\begin {gather*} \frac {b^2 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)}-\frac {\log (x) (a d+b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)}-\frac {1}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a c x^2}+\frac {-b c-a d}{a^2 c^2 x}-\frac {b^3}{a^2 (-b c+a d) (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a c x^2}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)}-\frac {d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 88, normalized size = 1.01 \begin {gather*} -\frac {1}{2 a c x^2}+\frac {(-b c-a d) \log (x)}{a^2 c^2}-\frac {b^2 \log \left (a+b x^2\right )}{2 a^2 (-b c+a d)}-\frac {d^2 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 83, normalized size = 0.95
method | result | size |
norman | \(-\frac {1}{2 c \,x^{2} a}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )}+\frac {d^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{2} \left (a d -b c \right )}-\frac {\left (a d +b c \right ) \ln \left (x \right )}{a^{2} c^{2}}\) | \(82\) |
default | \(-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )}+\frac {d^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{2} \left (a d -b c \right )}-\frac {1}{2 c \,x^{2} a}+\frac {\left (-a d -b c \right ) \ln \left (x \right )}{a^{2} c^{2}}\) | \(83\) |
risch | \(-\frac {1}{2 c \,x^{2} a}-\frac {\ln \left (x \right ) d}{a \,c^{2}}-\frac {\ln \left (x \right ) b}{a^{2} c}+\frac {d^{2} \ln \left (-d \,x^{2}-c \right )}{2 c^{2} \left (a d -b c \right )}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 87, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{2}} - \frac {1}{2 \, a c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.30, size = 99, normalized size = 1.14 \begin {gather*} \frac {b^{2} c^{2} x^{2} \log \left (b x^{2} + a\right ) - a^{2} d^{2} x^{2} \log \left (d x^{2} + c\right ) - a b c^{2} + a^{2} c d - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2} \log \left (x\right )}{2 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 112, normalized size = 1.29 \begin {gather*} \frac {b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{2} c - a^{3} b d\right )}} - \frac {d^{3} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c^{3} d - a c^{2} d^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{2}} + \frac {b c x^{2} + a d x^{2} - a c}{2 \, a^{2} c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {d^2\,\ln \left (d\,x^2+c\right )}{2\,\left (b\,c^3-a\,c^2\,d\right )}-\frac {1}{2\,a\,c\,x^2}-\frac {\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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