Optimal. Leaf size=86 \[ \frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac{c n}{6 b x^2}-\frac{n}{9 x^3} \]
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Rubi [A] time = 0.0571364, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 77} \[ \frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}-\frac{c n}{6 b x^2}-\frac{n}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 77
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x^4} \, dx &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \frac{b+2 c x}{x^4 (b+c x)} \, dx\\ &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \left (\frac{1}{x^4}+\frac{c}{b x^3}-\frac{c^2}{b^2 x^2}+\frac{c^3}{b^3 x}-\frac{c^4}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{n}{9 x^3}-\frac{c n}{6 b x^2}+\frac{c^2 n}{3 b^2 x}+\frac{c^3 n \log (x)}{3 b^3}-\frac{c^3 n \log (b+c x)}{3 b^3}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0352008, size = 77, normalized size = 0.9 \[ \frac{1}{3} n \left (\frac{c^2}{b^2 x}+\frac{c^3 \log (x)}{b^3}-\frac{c^3 \log (b+c x)}{b^3}-\frac{c}{2 b x^2}-\frac{1}{3 x^3}\right )-\frac{\log \left (d (x (b+c x))^n\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0167, size = 101, normalized size = 1.17 \begin{align*} -\frac{1}{18} \, n{\left (\frac{6 \, c^{3} \log \left (c x + b\right )}{b^{3}} - \frac{6 \, c^{3} \log \left (x\right )}{b^{3}} - \frac{6 \, c^{2} x^{2} - 3 \, b c x - 2 \, b^{2}}{b^{2} x^{3}}\right )} - \frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87479, size = 198, normalized size = 2.3 \begin{align*} -\frac{6 \, c^{3} n x^{3} \log \left (c x + b\right ) - 6 \, c^{3} n x^{3} \log \left (x\right ) - 6 \, b c^{2} n x^{2} + 3 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x^{2} + b x\right ) + 2 \, b^{3} n + 6 \, b^{3} \log \left (d\right )}{18 \, b^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.75475, size = 133, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{n \log{\left (b x + c x^{2} \right )}}{3 x^{3}} - \frac{n}{9 x^{3}} - \frac{\log{\left (d \right )}}{3 x^{3}} - \frac{c n}{6 b x^{2}} + \frac{c^{2} n}{3 b^{2} x} - \frac{2 c^{3} n \log{\left (b + c x \right )}}{3 b^{3}} + \frac{c^{3} n \log{\left (b x + c x^{2} \right )}}{3 b^{3}} & \text{for}\: b \neq 0 \\- \frac{n \log{\left (c \right )}}{3 x^{3}} - \frac{2 n \log{\left (x \right )}}{3 x^{3}} - \frac{2 n}{9 x^{3}} - \frac{\log{\left (d \right )}}{3 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16039, size = 108, normalized size = 1.26 \begin{align*} -\frac{c^{3} n \log \left (c x + b\right )}{3 \, b^{3}} + \frac{c^{3} n \log \left (x\right )}{3 \, b^{3}} - \frac{n \log \left (c x^{2} + b x\right )}{3 \, x^{3}} + \frac{6 \, c^{2} n x^{2} - 3 \, b c n x - 2 \, b^{2} n - 6 \, b^{2} \log \left (d\right )}{18 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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