Optimal. Leaf size=57 \[ -\frac {1}{(b c-a d) (a+b x)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 46}
\begin {gather*} -\frac {1}{(a+b x) (b c-a d)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 640
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \left (a c+(b c+a d) x+b d x^2\right )} \, dx &=\int \frac {1}{(a+b x)^2 (c+d x)} \, dx\\ &=\int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac {1}{(b c-a d) (a+b x)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.93 \begin {gather*} \frac {-b c+a d-d (a+b x) \log (a+b x)+d (a+b x) \log (c+d x)}{(b c-a d)^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 57, normalized size = 1.00
method | result | size |
default | \(\frac {d \ln \left (d x +c \right )}{\left (a d -b c \right )^{2}}+\frac {1}{\left (a d -b c \right ) \left (b x +a \right )}-\frac {d \ln \left (b x +a \right )}{\left (a d -b c \right )^{2}}\) | \(57\) |
risch | \(\frac {1}{\left (a d -b c \right ) \left (b x +a \right )}+\frac {d \ln \left (-d x -c \right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {d \ln \left (b x +a \right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}\) | \(86\) |
norman | \(-\frac {b x}{a \left (a d -b c \right ) \left (b x +a \right )}+\frac {d \ln \left (d x +c \right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {d \ln \left (b x +a \right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 92, normalized size = 1.61 \begin {gather*} -\frac {d \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac {d \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {1}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.62, size = 93, normalized size = 1.63 \begin {gather*} -\frac {b c - a d + {\left (b d x + a d\right )} \log \left (b x + a\right ) - {\left (b d x + a d\right )} \log \left (d x + c\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \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. 233 vs.
\(2 (46) = 92\).
time = 0.36, size = 233, normalized size = 4.09 \begin {gather*} \frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} + \frac {1}{a^{2} d - a b c + x \left (a b d - b^{2} c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 94, normalized size = 1.65 \begin {gather*} -\frac {b d \log \left ({\left | b x + a \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} - \frac {1}{{\left (b c - a d\right )} {\left (b x + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 76, normalized size = 1.33 \begin {gather*} \frac {1}{\left (a\,d-b\,c\right )\,\left (a+b\,x\right )}-\frac {2\,d\,\mathrm {atanh}\left (\frac {a^2\,d^2-b^2\,c^2}{{\left (a\,d-b\,c\right )}^2}+\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{{\left (a\,d-b\,c\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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