Optimal. Leaf size=140 \[ -\frac {p (b d-a e)^4 \log (a+b x)}{4 b^4 e}-\frac {p x (b d-a e)^3}{4 b^3}-\frac {p (d+e x)^2 (b d-a e)^2}{8 b^2 e}+\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e}-\frac {p (d+e x)^3 (b d-a e)}{12 b e}-\frac {p (d+e x)^4}{16 e} \]
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Rubi [A] time = 0.08, antiderivative size = 140, normalized size of antiderivative = 1.00, 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 = {2395, 43} \[ -\frac {p x (b d-a e)^3}{4 b^3}-\frac {p (d+e x)^2 (b d-a e)^2}{8 b^2 e}-\frac {p (b d-a e)^4 \log (a+b x)}{4 b^4 e}+\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e}-\frac {p (d+e x)^3 (b d-a e)}{12 b e}-\frac {p (d+e x)^4}{16 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rubi steps
\begin {align*} \int (d+e x)^3 \log \left (c (a+b x)^p\right ) \, dx &=\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e}-\frac {(b p) \int \frac {(d+e x)^4}{a+b x} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e}-\frac {(b p) \int \left (\frac {e (b d-a e)^3}{b^4}+\frac {(b d-a e)^4}{b^4 (a+b x)}+\frac {e (b d-a e)^2 (d+e x)}{b^3}+\frac {e (b d-a e) (d+e x)^2}{b^2}+\frac {e (d+e x)^3}{b}\right ) \, dx}{4 e}\\ &=-\frac {(b d-a e)^3 p x}{4 b^3}-\frac {(b d-a e)^2 p (d+e x)^2}{8 b^2 e}-\frac {(b d-a e) p (d+e x)^3}{12 b e}-\frac {p (d+e x)^4}{16 e}-\frac {(b d-a e)^4 p \log (a+b x)}{4 b^4 e}+\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 185, normalized size = 1.32 \[ -\frac {12 a^2 e p \left (a^2 e^2-4 a b d e+6 b^2 d^2\right ) \log (a+b x)+b p x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )-12 b^3 \left (4 a d^3+b x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \log \left (c (a+b x)^p\right )}{48 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 269, normalized size = 1.92 \[ -\frac {3 \, b^{4} e^{3} p x^{4} + 4 \, {\left (4 \, b^{4} d e^{2} - a b^{3} e^{3}\right )} p x^{3} + 6 \, {\left (6 \, b^{4} d^{2} e - 4 \, a b^{3} d e^{2} + a^{2} b^{2} e^{3}\right )} p x^{2} + 12 \, {\left (4 \, b^{4} d^{3} - 6 \, a b^{3} d^{2} e + 4 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} p x - 12 \, {\left (b^{4} e^{3} p x^{4} + 4 \, b^{4} d e^{2} p x^{3} + 6 \, b^{4} d^{2} e p x^{2} + 4 \, b^{4} d^{3} p x + {\left (4 \, a b^{3} d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a^{3} b d e^{2} - a^{4} e^{3}\right )} p\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} e^{3} x^{4} + 4 \, b^{4} d e^{2} x^{3} + 6 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} d^{3} x\right )} \log \relax (c)}{48 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 558, normalized size = 3.99 \[ \frac {{\left (b x + a\right )} d^{3} p \log \left (b x + a\right )}{b} + \frac {3 \, {\left (b x + a\right )}^{2} d^{2} p e \log \left (b x + a\right )}{2 \, b^{2}} - \frac {3 \, {\left (b x + a\right )} a d^{2} p e \log \left (b x + a\right )}{b^{2}} - \frac {{\left (b x + a\right )} d^{3} p}{b} - \frac {3 \, {\left (b x + a\right )}^{2} d^{2} p e}{4 \, b^{2}} + \frac {3 \, {\left (b x + a\right )} a d^{2} p e}{b^{2}} + \frac {{\left (b x + a\right )}^{3} d p e^{2} \log \left (b x + a\right )}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a d p e^{2} \log \left (b x + a\right )}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} d p e^{2} \log \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} d^{3} \log \relax (c)}{b} + \frac {3 \, {\left (b x + a\right )}^{2} d^{2} e \log \relax (c)}{2 \, b^{2}} - \frac {3 \, {\left (b x + a\right )} a d^{2} e \log \relax (c)}{b^{2}} - \frac {{\left (b x + a\right )}^{3} d p e^{2}}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a d p e^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x + a\right )} a^{2} d p e^{2}}{b^{3}} + \frac {{\left (b x + a\right )}^{4} p e^{3} \log \left (b x + a\right )}{4 \, b^{4}} - \frac {{\left (b x + a\right )}^{3} a p e^{3} \log \left (b x + a\right )}{b^{4}} + \frac {3 \, {\left (b x + a\right )}^{2} a^{2} p e^{3} \log \left (b x + a\right )}{2 \, b^{4}} - \frac {{\left (b x + a\right )} a^{3} p e^{3} \log \left (b x + a\right )}{b^{4}} + \frac {{\left (b x + a\right )}^{3} d e^{2} \log \relax (c)}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a d e^{2} \log \relax (c)}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} d e^{2} \log \relax (c)}{b^{3}} - \frac {{\left (b x + a\right )}^{4} p e^{3}}{16 \, b^{4}} + \frac {{\left (b x + a\right )}^{3} a p e^{3}}{3 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{2} a^{2} p e^{3}}{4 \, b^{4}} + \frac {{\left (b x + a\right )} a^{3} p e^{3}}{b^{4}} + \frac {{\left (b x + a\right )}^{4} e^{3} \log \relax (c)}{4 \, b^{4}} - \frac {{\left (b x + a\right )}^{3} a e^{3} \log \relax (c)}{b^{4}} + \frac {3 \, {\left (b x + a\right )}^{2} a^{2} e^{3} \log \relax (c)}{2 \, b^{4}} - \frac {{\left (b x + a\right )} a^{3} e^{3} \log \relax (c)}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 766, normalized size = 5.47 \[ d \,e^{2} x^{3} \ln \relax (c )+\frac {3 d^{2} e \,x^{2} \ln \relax (c )}{2}-\frac {d^{4} p \ln \left (b x +a \right )}{4 e}+\frac {e^{3} x^{4} \ln \relax (c )}{4}+d^{3} x \ln \relax (c )+\frac {\left (e x +d \right )^{4} \ln \left (\left (b x +a \right )^{p}\right )}{4 e}-\frac {e^{3} p \,x^{4}}{16}-\frac {3 d^{2} e p \,x^{2}}{4}+\frac {a \,e^{3} p \,x^{3}}{12 b}-\frac {a^{2} e^{3} p \,x^{2}}{8 b^{2}}+\frac {a^{3} e^{3} p x}{4 b^{3}}-d^{3} p x -\frac {a^{4} e^{3} p \ln \left (b x +a \right )}{4 b^{4}}+\frac {a \,d^{3} p \ln \left (b x +a \right )}{b}-\frac {d \,e^{2} p \,x^{3}}{3}+\frac {a^{3} d \,e^{2} p \ln \left (b x +a \right )}{b^{3}}-\frac {3 a^{2} d^{2} e p \ln \left (b x +a \right )}{2 b^{2}}+\frac {i \pi \,e^{3} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{8}+\frac {i \pi \,e^{3} x^{4} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{8}-\frac {i \pi d \,e^{2} x^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}-\frac {3 i \pi \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{4}+\frac {i \pi \,d^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}+\frac {i \pi \,d^{3} x \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi d \,e^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{2}-\frac {3 i \pi \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{4}+\frac {a d \,e^{2} p \,x^{2}}{2 b}-\frac {a^{2} d \,e^{2} p x}{b^{2}}+\frac {3 a \,d^{2} e p x}{2 b}-\frac {i \pi \,e^{3} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{8}-\frac {i \pi \,e^{3} x^{4} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{8}-\frac {i \pi \,d^{3} x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi d \,e^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}+\frac {i \pi d \,e^{2} x^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}+\frac {3 i \pi \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{4}+\frac {3 i \pi \,d^{2} e \,x^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{4}-\frac {i \pi \,d^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 214, normalized size = 1.53 \[ -\frac {1}{48} \, b p {\left (\frac {3 \, b^{3} e^{3} x^{4} + 4 \, {\left (4 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{3} + 6 \, {\left (6 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + 12 \, {\left (4 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 4 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} x}{b^{4}} - \frac {12 \, {\left (4 \, a b^{3} d^{3} - 6 \, a^{2} b^{2} d^{2} e + 4 \, a^{3} b d e^{2} - a^{4} e^{3}\right )} \log \left (b x + a\right )}{b^{5}}\right )} + \frac {1}{4} \, {\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 208, normalized size = 1.49 \[ \ln \left (c\,{\left (a+b\,x\right )}^p\right )\,\left (d^3\,x+\frac {3\,d^2\,e\,x^2}{2}+d\,e^2\,x^3+\frac {e^3\,x^4}{4}\right )+x^2\,\left (\frac {a\,\left (d\,e^2\,p-\frac {a\,e^3\,p}{4\,b}\right )}{2\,b}-\frac {3\,d^2\,e\,p}{4}\right )-x\,\left (d^3\,p+\frac {a\,\left (\frac {a\,\left (d\,e^2\,p-\frac {a\,e^3\,p}{4\,b}\right )}{b}-\frac {3\,d^2\,e\,p}{2}\right )}{b}\right )-x^3\,\left (\frac {d\,e^2\,p}{3}-\frac {a\,e^3\,p}{12\,b}\right )-\frac {e^3\,p\,x^4}{16}-\frac {\ln \left (a+b\,x\right )\,\left (p\,a^4\,e^3-4\,p\,a^3\,b\,d\,e^2+6\,p\,a^2\,b^2\,d^2\,e-4\,p\,a\,b^3\,d^3\right )}{4\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.40, size = 369, normalized size = 2.64 \[ \begin {cases} - \frac {a^{4} e^{3} p \log {\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} d e^{2} p \log {\left (a + b x \right )}}{b^{3}} + \frac {a^{3} e^{3} p x}{4 b^{3}} - \frac {3 a^{2} d^{2} e p \log {\left (a + b x \right )}}{2 b^{2}} - \frac {a^{2} d e^{2} p x}{b^{2}} - \frac {a^{2} e^{3} p x^{2}}{8 b^{2}} + \frac {a d^{3} p \log {\left (a + b x \right )}}{b} + \frac {3 a d^{2} e p x}{2 b} + \frac {a d e^{2} p x^{2}}{2 b} + \frac {a e^{3} p x^{3}}{12 b} + d^{3} p x \log {\left (a + b x \right )} - d^{3} p x + d^{3} x \log {\relax (c )} + \frac {3 d^{2} e p x^{2} \log {\left (a + b x \right )}}{2} - \frac {3 d^{2} e p x^{2}}{4} + \frac {3 d^{2} e x^{2} \log {\relax (c )}}{2} + d e^{2} p x^{3} \log {\left (a + b x \right )} - \frac {d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log {\relax (c )} + \frac {e^{3} p x^{4} \log {\left (a + b x \right )}}{4} - \frac {e^{3} p x^{4}}{16} + \frac {e^{3} x^{4} \log {\relax (c )}}{4} & \text {for}\: b \neq 0 \\\left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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