Optimal. Leaf size=57 \[ \frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B n (b c-a d) \log (c+d x)}{b d}+A x \]
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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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2486, 31} \[ \frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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Rule 31
Rule 2486
Rubi steps
\begin {align*} \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=A x+B \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=A x+\frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {(B (b c-a d) n) \int \frac {1}{c+d x} \, dx}{b}\\ &=A x-\frac {B (b c-a d) n \log (c+d x)}{b d}+\frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 1.00 \[ \frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B n (b c-a d) \log (c+d x)}{b d}+A x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 59, normalized size = 1.04 \[ \frac {B b d x \log \relax (e) + A b d x + {\left (B b d n x + B a d n\right )} \log \left (b x + a\right ) - {\left (B b d n x + B b c n\right )} \log \left (d x + c\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 55, normalized size = 0.96 \[ {\left (n x \log \left (b x + a\right ) - n x \log \left (d x + c\right ) + \frac {a n \log \left (b x + a\right )}{b} - \frac {c n \log \left (-d x - c\right )}{d} + x\right )} B + A x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 123, normalized size = 2.16 \[ \frac {B \,a^{2} d n \ln \left (b x +a \right )}{\left (a d -b c \right ) b}-\frac {B a c n \ln \left (b x +a \right )}{a d -b c}-\frac {B a c n \ln \left (d x +c \right )}{a d -b c}+\frac {B b \,c^{2} n \ln \left (d x +c \right )}{\left (a d -b c \right ) d}+B x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 59, normalized size = 1.04 \[ B x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A x + \frac {{\left (\frac {a e n \log \left (b x + a\right )}{b} - \frac {c e n \log \left (d x + c\right )}{d}\right )} B}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.11, size = 53, normalized size = 0.93 \[ A\,x+B\,x\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )+\frac {B\,a\,n\,\ln \left (a+b\,x\right )}{b}-\frac {B\,c\,n\,\ln \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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