Optimal. Leaf size=56 \[ \frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)} \]
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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84}
\begin {gather*} \frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac {x}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x) (c+d x)} \, dx &=\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\\ &=\frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 1.00 \begin {gather*} \frac {x}{b d}+\frac {a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac {c^2 \log (c+d x)}{d^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 57, normalized size = 1.02
method | result | size |
default | \(\frac {x}{b d}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (a d -b c \right )}+\frac {c^{2} \ln \left (d x +c \right )}{d^{2} \left (a d -b c \right )}\) | \(57\) |
norman | \(\frac {x}{b d}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (a d -b c \right )}+\frac {c^{2} \ln \left (d x +c \right )}{d^{2} \left (a d -b c \right )}\) | \(57\) |
risch | \(\frac {x}{b d}+\frac {c^{2} \ln \left (-d x -c \right )}{d^{2} \left (a d -b c \right )}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (a d -b c \right )}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 60, normalized size = 1.07 \begin {gather*} \frac {a^{2} \log \left (b x + a\right )}{b^{3} c - a b^{2} d} - \frac {c^{2} \log \left (d x + c\right )}{b c d^{2} - a d^{3}} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 65, normalized size = 1.16 \begin {gather*} \frac {a^{2} d^{2} \log \left (b x + a\right ) - b^{2} c^{2} \log \left (d x + c\right ) + {\left (b^{2} c d - a b d^{2}\right )} x}{b^{3} c d^{2} - a b^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs.
\(2 (44) = 88\).
time = 0.60, size = 190, normalized size = 3.39 \begin {gather*} - \frac {a^{2} \log {\left (x + \frac {\frac {a^{4} d^{3}}{b \left (a d - b c\right )} - \frac {2 a^{3} c d^{2}}{a d - b c} + \frac {a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{b^{2} \left (a d - b c\right )} + \frac {c^{2} \log {\left (x + \frac {- \frac {a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac {2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac {b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 62, normalized size = 1.11 \begin {gather*} \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} c - a b^{2} d} - \frac {c^{2} \log \left ({\left | d x + c \right |}\right )}{b c d^{2} - a d^{3}} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.44, size = 61, normalized size = 1.09 \begin {gather*} -\frac {a^2\,d^2\,\ln \left (a+b\,x\right )-b^2\,c^2\,\ln \left (c+d\,x\right )-a\,b\,d^2\,x+b^2\,c\,d\,x}{b^2\,d^2\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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