Optimal. Leaf size=242 \[ -\frac {B d i^2 n (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 n (c+d x)^2}{4 b g^3 (a+b x)^2}-\frac {d i^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}+\frac {B d^2 i^2 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3} \]
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Rubi [A]
time = 0.22, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2561, 2380,
2341, 2379, 2438} \begin {gather*} \frac {B d^2 i^2 n \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac {d^2 i^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^3}-\frac {d i^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac {B d i^2 n (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 n (c+d x)^2}{4 b g^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2379
Rule 2380
Rule 2438
Rule 2561
Rubi steps
\begin {align*} \int \frac {(123 c+123 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)^3}+\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)^2}+\frac {15129 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (15129 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^3}+\frac {(30258 d (b c-a d)) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac {\left (15129 (b c-a d)^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\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}{b^3 g^3}+\frac {(30258 B d (b c-a d) n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (15129 B (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^3}\\ &=-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^3}+\frac {\left (30258 B d (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (15129 B (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^3}\\ &=-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (15129 B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^3}+\frac {\left (30258 B d (b c-a d)^2 n\right ) \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}{b^3 g^3}+\frac {\left (15129 B (b c-a d)^3 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^3 g^3}\\ &=-\frac {15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac {45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac {45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac {45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac {15129 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}\\ &=-\frac {15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac {45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac {45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac {15129 B d^2 n \log ^2(a+b x)}{2 b^3 g^3}-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac {45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac {15129 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (15129 B d^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^3}\\ &=-\frac {15129 B (b c-a d)^2 n}{4 b^3 g^3 (a+b x)^2}-\frac {45387 B d (b c-a d) n}{2 b^3 g^3 (a+b x)}-\frac {45387 B d^2 n \log (a+b x)}{2 b^3 g^3}-\frac {15129 B d^2 n \log ^2(a+b x)}{2 b^3 g^3}-\frac {15129 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^3 (a+b x)^2}-\frac {30258 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}+\frac {15129 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac {45387 B d^2 n \log (c+d x)}{2 b^3 g^3}+\frac {15129 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {15129 B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 258, normalized size = 1.07 \begin {gather*} \frac {i^2 \left (-\frac {B (b c-a d)^2 n}{(a+b x)^2}+\frac {6 B d (-b c+a d) n}{a+b x}-6 B d^2 n \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {8 d (-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}+4 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 B d^2 n \log (c+d x)-2 B d^2 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 )\right )}{4 b^3 g^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (b g x +a g \right )^{3}}\, dx\]
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 [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*} \frac {i^{2} \left (\int \frac {A c^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {A d^{2} x^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A c d x}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 B c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx\right )}{g^{3}} \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 )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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