Optimal. Leaf size=206 \[ \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b f-a g) (f+g x)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g) (d f-c g)}+\frac {2 B^2 (b c-a d) n^2 \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g) (d f-c g)} \]
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Rubi [A]
time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2553, 2355,
2354, 2438} \begin {gather*} \frac {2 B^2 n^2 (b c-a d) \text {PolyLog}\left (2,\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{(b f-a g) (d f-c g)}+\frac {2 B n (b c-a d) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(b f-a g) (d f-c g)}+\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(f+g x) (b f-a g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2355
Rule 2438
Rule 2553
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^2} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {(2 B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {(2 B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {(2 B (b c-a d) n) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (b f-a g) (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (-d f+c g) (c+d x)}+\frac {g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g) (d f-c g) (f+g x)}\right ) \, dx}{g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {\left (2 b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{g (b f-a g)}-\frac {\left (2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{g (d f-c g)}+\frac {(2 B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (2 b B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{g (b f-a g)}+\frac {\left (2 B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{g (d f-c g)}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (f+g x)}{a+b x} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (2 b B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{g (b f-a g)}+\frac {\left (2 B^2 d n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{g (d f-c g)}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac {b \log (f+g x)}{a+b x}-\frac {d \log (f+g x)}{c+d x}\right ) \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{g (b f-a g)}+\frac {\left (2 b B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{g (b f-a g)}+\frac {\left (2 b B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{g (d f-c g)}-\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{g (d f-c g)}-\frac {\left (2 b B^2 (b c-a d) n^2\right ) \int \frac {\log (f+g x)}{a+b x} \, dx}{(b f-a g) (d f-c g)}+\frac {\left (2 B^2 d (b c-a d) n^2\right ) \int \frac {\log (f+g x)}{c+d x} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (2 b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{g (b f-a g)}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{g (b f-a g)}-\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{g (d f-c g)}-\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{g (d f-c g)}+\frac {\left (2 B^2 (b c-a d) g n^2\right ) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}-\frac {\left (2 B^2 (b c-a d) g n^2\right ) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}\\ &=-\frac {b B^2 n^2 \log ^2(a+b x)}{g (b f-a g)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}-\frac {B^2 d n^2 \log ^2(c+d x)}{g (d f-c g)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (2 b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{g (b f-a g)}-\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{g (d f-c g)}+\frac {\left (2 B^2 (b c-a d) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g) (d f-c g)}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g) (d f-c g)}\\ &=-\frac {b B^2 n^2 \log ^2(a+b x)}{g (b f-a g)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{g (f+g x)}+\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)}-\frac {B^2 d n^2 \log ^2(c+d x)}{g (d f-c g)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 B^2 (b c-a d) n^2 \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {2 b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g (b f-a g)}+\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g (d f-c g)}-\frac {2 B^2 (b c-a d) n^2 \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{(b f-a g) (d f-c g)}+\frac {2 B^2 (b c-a d) n^2 \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{(b f-a g) (d f-c g)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(206)=412\).
time = 0.34, size = 418, normalized size = 2.03 \begin {gather*} \frac {-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{f+g x}+\frac {B n \left (2 b (d f-c g) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 d (b f-a g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+2 (b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)-b B (d f-c g) n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (b f-a g) n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) g n \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )-\text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g) (d f-c g)}}{g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (g x +f \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\right )^{2}}{\left (f + g x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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