Optimal. Leaf size=129 \[ -\frac {2 B n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)}-\frac {2 B^2 n^2 (c+d x)}{(a+b x) (b c-a d)} \]
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Rubi [A] time = 0.18, antiderivative size = 189, normalized size of antiderivative = 1.47, number of steps used = 7, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6742, 2490, 32} \[ -\frac {A^2}{b (a+b x)}-\frac {2 A B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {2 A B n}{b (a+b x)}-\frac {B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {2 B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {2 B^2 n^2}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2490
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^2} \, dx &=\int \left (\frac {A^2}{(a+b x)^2}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}\right ) \, dx\\ &=-\frac {A^2}{b (a+b x)}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx\\ &=-\frac {A^2}{b (a+b x)}-\frac {2 A B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}+(2 A B n) \int \frac {1}{(a+b x)^2} \, dx+\left (2 B^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx\\ &=-\frac {A^2}{b (a+b x)}-\frac {2 A B n}{b (a+b x)}-\frac {2 A B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {2 B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}+\left (2 B^2 n^2\right ) \int \frac {1}{(a+b x)^2} \, dx\\ &=-\frac {A^2}{b (a+b x)}-\frac {2 A B n}{b (a+b x)}-\frac {2 B^2 n^2}{b (a+b x)}-\frac {2 A B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {2 B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}-\frac {B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 236, normalized size = 1.83 \[ \frac {-(b c-a d) \left (2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2+2 A B n+2 B^2 n^2\right )-2 B d n (a+b x) \log (a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n \log (c+d x)+B n\right )+2 B d n (a+b x) \log (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )+B^2 d n^2 (a+b x) \log ^2(c+d x)+B^2 d n^2 (a+b x) \log ^2(a+b x)}{b (a+b x) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 339, normalized size = 2.63 \[ -\frac {A^{2} b c - A^{2} a d + 2 \, {\left (B^{2} b c - B^{2} a d\right )} n^{2} + {\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (b x + a\right )^{2} + {\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (d x + c\right )^{2} + {\left (B^{2} b c - B^{2} a d\right )} \log \relax (e)^{2} + 2 \, {\left (A B b c - A B a d\right )} n + 2 \, {\left (B^{2} b c n^{2} + A B b c n + {\left (B^{2} b d n^{2} + A B b d n\right )} x + {\left (B^{2} b d n x + B^{2} b c n\right )} \log \relax (e)\right )} \log \left (b x + a\right ) - 2 \, {\left (B^{2} b c n^{2} + A B b c n + {\left (B^{2} b d n^{2} + A B b d n\right )} x + {\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (b x + a\right ) + {\left (B^{2} b d n x + B^{2} b c n\right )} \log \relax (e)\right )} \log \left (d x + c\right ) + 2 \, {\left (A B b c - A B a d + {\left (B^{2} b c - B^{2} a d\right )} n\right )} \log \relax (e)}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \]
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 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.25, size = 10098, normalized size = 78.28 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 449, normalized size = 3.48 \[ -B^{2} {\left (\frac {2 \, {\left (\frac {d e n \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d e n \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {e n}{b^{2} x + a b}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{e} + \frac {2 \, b c e^{2} n^{2} - 2 \, a d e^{2} n^{2} - {\left (b d e^{2} n^{2} x + a d e^{2} n^{2}\right )} \log \left (b x + a\right )^{2} - {\left (b d e^{2} n^{2} x + a d e^{2} n^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left (b d e^{2} n^{2} x + a d e^{2} n^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (b d e^{2} n^{2} x + a d e^{2} n^{2} - {\left (b d e^{2} n^{2} x + a d e^{2} n^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{{\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x\right )} e^{2}}\right )} - \frac {B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{b^{2} x + a b} - \frac {2 \, {\left (\frac {d e n \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d e n \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {e n}{b^{2} x + a b}\right )} A B}{e} - \frac {2 \, A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{b^{2} x + a b} - \frac {A^{2}}{b^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 200, normalized size = 1.55 \[ -\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {2\,A\,B}{x\,b^2+a\,b}+\frac {2\,B^2\,n}{x\,b^2+a\,b}\right )-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {B^2}{b\,\left (a+b\,x\right )}-\frac {B^2\,d}{b\,\left (a\,d-b\,c\right )}\right )-\frac {A^2+2\,A\,B\,n+2\,B^2\,n^2}{x\,b^2+a\,b}-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {\left (\frac {c\,b^2+a\,d\,b}{b}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+B\,n\right )\,4{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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