Optimal. Leaf size=139 \[ \frac {b e \left (6 a^2 d^2-4 a b d e+b^2 e^2\right ) p x}{4 a^3}+\frac {b e^2 (4 a d-b e) p x^2}{8 a^2}+\frac {b e^3 p x^3}{12 a}+\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {d^4 p \log (x)}{4 e}-\frac {(a d-b e)^4 p \log (b+a x)}{4 a^4 e} \]
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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 528, 84}
\begin {gather*} -\frac {p (a d-b e)^4 \log (a x+b)}{4 a^4 e}+\frac {b e^2 p x^2 (4 a d-b e)}{8 a^2}+\frac {b e p x \left (6 a^2 d^2-4 a b d e+b^2 e^2\right )}{4 a^3}+\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {b e^3 p x^3}{12 a}+\frac {d^4 p \log (x)}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 528
Rule 2513
Rubi steps
\begin {align*} \int (d+e x)^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx &=\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {(b p) \int \frac {(d+e x)^4}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {(b p) \int \frac {(d+e x)^4}{x (b+a x)} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {(b p) \int \left (\frac {e^2 \left (6 a^2 d^2-4 a b d e+b^2 e^2\right )}{a^3}+\frac {d^4}{b x}+\frac {e^3 (4 a d-b e) x}{a^2}+\frac {e^4 x^2}{a}-\frac {(a d-b e)^4}{a^3 b (b+a x)}\right ) \, dx}{4 e}\\ &=\frac {b e \left (6 a^2 d^2-4 a b d e+b^2 e^2\right ) p x}{4 a^3}+\frac {b e^2 (4 a d-b e) p x^2}{8 a^2}+\frac {b e^3 p x^3}{12 a}+\frac {(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 e}+\frac {d^4 p \log (x)}{4 e}-\frac {(a d-b e)^4 p \log (b+a x)}{4 a^4 e}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 114, normalized size = 0.82 \begin {gather*} \frac {\frac {b e^2 p x \left (6 b^2 e^2-3 a b e (8 d+e x)+2 a^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )}{6 a^3}+(d+e x)^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )+d^4 p \log (x)-\frac {(a d-b e)^4 p \log (b+a x)}{a^4}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 160, normalized size = 1.15 \begin {gather*} \frac {1}{24} \, b p {\left (\frac {2 \, a^{2} x^{3} e^{3} + 3 \, {\left (4 \, a^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \, {\left (6 \, a^{2} d^{2} e - 4 \, a b d e^{2} + b^{2} e^{3}\right )} x}{a^{3}} + \frac {6 \, {\left (4 \, a^{3} d^{3} - 6 \, a^{2} b d^{2} e + 4 \, a b^{2} d e^{2} - b^{3} e^{3}\right )} \log \left (a x + b\right )}{a^{4}}\right )} + \frac {1}{4} \, {\left (x^{4} e^{3} + 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e + 4 \, d^{3} x\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 236, normalized size = 1.70 \begin {gather*} \frac {36 \, a^{3} b d^{2} p x e + {\left (2 \, a^{3} b p x^{3} - 3 \, a^{2} b^{2} p x^{2} + 6 \, a b^{3} p x\right )} e^{3} + 12 \, {\left (a^{3} b d p x^{2} - 2 \, a^{2} b^{2} d p x\right )} e^{2} + 6 \, {\left (4 \, a^{3} b d^{3} p - 6 \, a^{2} b^{2} d^{2} p e + 4 \, a b^{3} d p e^{2} - b^{4} p e^{3}\right )} \log \left (a x + b\right ) + 6 \, {\left (a^{4} x^{4} e^{3} + 4 \, a^{4} d x^{3} e^{2} + 6 \, a^{4} d^{2} x^{2} e + 4 \, a^{4} d^{3} x\right )} \log \left (c\right ) + 6 \, {\left (a^{4} p x^{4} e^{3} + 4 \, a^{4} d p x^{3} e^{2} + 6 \, a^{4} d^{2} p x^{2} e + 4 \, a^{4} d^{3} p x\right )} \log \left (\frac {a x + b}{x}\right )}{24 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs.
\(2 (128) = 256\).
time = 1.71, size = 355, normalized size = 2.55 \begin {gather*} \begin {cases} d^{3} x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {3 d^{2} e x^{2} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{2} + d e^{2} x^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {e^{3} x^{4} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{4} + \frac {b d^{3} p \log {\left (x + \frac {b}{a} \right )}}{a} + \frac {3 b d^{2} e p x}{2 a} + \frac {b d e^{2} p x^{2}}{2 a} + \frac {b e^{3} p x^{3}}{12 a} - \frac {3 b^{2} d^{2} e p \log {\left (x + \frac {b}{a} \right )}}{2 a^{2}} - \frac {b^{2} d e^{2} p x}{a^{2}} - \frac {b^{2} e^{3} p x^{2}}{8 a^{2}} + \frac {b^{3} d e^{2} p \log {\left (x + \frac {b}{a} \right )}}{a^{3}} + \frac {b^{3} e^{3} p x}{4 a^{3}} - \frac {b^{4} e^{3} p \log {\left (x + \frac {b}{a} \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\d^{3} p x + d^{3} x \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {3 d^{2} e p x^{2}}{4} + \frac {3 d^{2} e x^{2} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{2} + \frac {d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} + \frac {e^{3} p x^{4}}{16} + \frac {e^{3} x^{4} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1659 vs.
\(2 (126) = 252\).
time = 5.02, size = 1659, normalized size = 11.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 184, normalized size = 1.32 \begin {gather*} x\,\left (\frac {b\,\left (\frac {b^2\,e^3\,p}{4\,a^2}-\frac {b\,d\,e^2\,p}{a}\right )}{a}+\frac {3\,b\,d^2\,e\,p}{2\,a}\right )+\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )\,\left (d^3\,x+\frac {3\,d^2\,e\,x^2}{2}+d\,e^2\,x^3+\frac {e^3\,x^4}{4}\right )-x^2\,\left (\frac {b^2\,e^3\,p}{8\,a^2}-\frac {b\,d\,e^2\,p}{2\,a}\right )-\frac {\ln \left (b+a\,x\right )\,\left (-4\,p\,a^3\,b\,d^3+6\,p\,a^2\,b^2\,d^2\,e-4\,p\,a\,b^3\,d\,e^2+p\,b^4\,e^3\right )}{4\,a^4}+\frac {b\,e^3\,p\,x^3}{12\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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