Optimal. Leaf size=309 \[ -\frac {6 B^3 n^3 (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{(b c-a d) i^2 (1+m)^4 (c+d x)}-\frac {6 B^2 n^2 (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 (1+m)^3 (c+d x)}-\frac {3 B n (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d) i^2 (1+m)^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(b c-a d) i^2 (1+m) (c+d x)} \]
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Rubi [A]
time = 0.22, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2563, 2342,
2341} \begin {gather*} -\frac {6 B^2 n^2 (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1)^3 (c+d x) (b c-a d)}-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{i^2 (m+1) (c+d x) (b c-a d)}-\frac {3 B n (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (m+1)^2 (c+d x) (b c-a d)}-\frac {6 B^3 n^3 (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{i^2 (m+1)^4 (c+d x) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
Rule 2563
Rubi steps
\begin {align*} \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 \, dx &=\int \left (A^3 (218 c+218 d x)^m (a g+b g x)^{-2-m}+3 A^2 B (218 c+218 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A B^2 (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^3 (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=A^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \, dx+\left (3 A^2 B\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac {A^3 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{218 (b c-a d) g (1+m)}-\left (3 A^2 B\right ) \int \frac {218^m n (c+d x)^m (a g+b g x)^{-2-m}}{-1-m} \, dx+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac {A^3 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{218 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+\frac {\left (3\ 218^m A^2 B n\right ) \int (c+d x)^m (a g+b g x)^{-2-m} \, dx}{1+m}\\ &=-\frac {3\ 218^m A^2 B n (c+d x)^{1+m} (a g+b g x)^{-1-m}}{(b c-a d) g (1+m)^2}-\frac {A^3 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{218 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ \end {align*}
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Mathematica [A]
time = 4.38, size = 206, normalized size = 0.67 \begin {gather*} -\frac {(g (a+b x))^{-1-m} (c+d x) (i (c+d x))^m \left (A^3 (1+m)^3+3 A^2 B (1+m)^2 n+6 A B^2 (1+m) n^2+6 B^3 n^3+3 B (1+m) \left (A^2 (1+m)^2+2 A B (1+m) n+2 B^2 n^2\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 B^2 (1+m)^2 (A+A m+B n) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^3 (1+m)^3 \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) g (1+m)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{-2-m} \left (d i x +c i \right )^{m} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{3}\, dx\]
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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2059 vs. \(2 (303) = 606\).
time = 0.43, size = 2059, normalized size = 6.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^3}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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