Optimal. Leaf size=70 \[ \frac {x^2}{2 b d}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac {c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, 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 {a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac {c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}+\frac {x^2}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b d}+\frac {a^2 \log \left (a+b x^2\right )}{2 b^2 (b c-a d)}-\frac {c^2 \log \left (c+d x^2\right )}{2 d^2 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 66, normalized size = 0.94 \begin {gather*} \frac {a^2 d^2 \log \left (a+b x^2\right )-b \left (d (-b c+a d) x^2+b c^2 \log \left (c+d x^2\right )\right )}{2 b^2 d^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 65, normalized size = 0.93
method | result | size |
default | \(\frac {x^{2}}{2 b d}-\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{2} \left (a d -b c \right )}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 d^{2} \left (a d -b c \right )}\) | \(65\) |
norman | \(\frac {x^{2}}{2 b d}-\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{2} \left (a d -b c \right )}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 d^{2} \left (a d -b c \right )}\) | \(65\) |
risch | \(\frac {x^{2}}{2 b d}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 d^{2} \left (a d -b c \right )}-\frac {a^{2} \ln \left (-b \,x^{2}-a \right )}{2 b^{2} \left (a d -b c \right )}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 68, normalized size = 0.97 \begin {gather*} \frac {a^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}} - \frac {c^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b c d^{2} - a d^{3}\right )}} + \frac {x^{2}}{2 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 72, normalized size = 1.03 \begin {gather*} \frac {a^{2} d^{2} \log \left (b x^{2} + a\right ) - b^{2} c^{2} \log \left (d x^{2} + c\right ) + {\left (b^{2} c d - a b d^{2}\right )} x^{2}}{2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}} \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 = 0.57, size = 70, normalized size = 1.00 \begin {gather*} \frac {a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}} - \frac {c^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c d^{2} - a d^{3}\right )}} + \frac {x^{2}}{2 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 68, normalized size = 0.97 \begin {gather*} \frac {a^2\,\ln \left (b\,x^2+a\right )}{2\,b^3\,c-2\,a\,b^2\,d}+\frac {c^2\,\ln \left (d\,x^2+c\right )}{2\,a\,d^3-2\,b\,c\,d^2}+\frac {x^2}{2\,b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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