Optimal. Leaf size=140 \[ -\frac {(b d-a e)^3 p x}{4 b^3}-\frac {(b d-a e)^2 p (d+e x)^2}{8 b^2 e}-\frac {(b d-a e) p (d+e x)^3}{12 b e}-\frac {p (d+e x)^4}{16 e}-\frac {(b d-a e)^4 p \log (a+b x)}{4 b^4 e}+\frac {(d+e x)^4 \log \left (c (a+b x)^p\right )}{4 e} \]
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Rubi [A]
time = 0.05, 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 = {2442, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
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 = 0.14, size = 185, normalized size = 1.32 \begin {gather*} -\frac {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 a^2 e \left (6 b^2 d^2-4 a b d e+a^2 e^2\right ) p \log (a+b x)-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} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.72, size = 766, normalized size = 5.47
method | result | size |
risch | \(\frac {e^{3} a p \,x^{3}}{12 b}-\frac {e^{3} a^{2} p \,x^{2}}{8 b^{2}}-d^{3} p x +\frac {d^{3} p a \ln \left (b x +a \right )}{b}+e^{2} \ln \left (c \right ) d \,x^{3}+\frac {3 e \ln \left (c \right ) d^{2} x^{2}}{2}-\frac {\ln \left (b x +a \right ) d^{4} p}{4 e}-\frac {e^{2} a^{2} d p x}{b^{2}}+\frac {3 e a \,d^{2} p x}{2 b}+\frac {e^{2} \ln \left (b x +a \right ) a^{3} d p}{b^{3}}-\frac {3 i e \pi \,d^{2} x^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{4}+\frac {i e^{3} \pi \,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 e^{3} \pi \,x^{4} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{8}-\frac {i e^{2} \pi d \,x^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}-\frac {3 i e \pi \,d^{2} 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 ) \mathrm {csgn}\left (i c \right )}{4}+\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 {e^{3} \ln \left (c \right ) x^{4}}{4}+\ln \left (c \right ) d^{3} x -\frac {i \pi \,d^{3} x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}-\frac {i e^{3} \pi \,x^{4} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{8}-\frac {e^{3} p \,x^{4}}{16}-\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 ) \mathrm {csgn}\left (i c \right )}{2}-\frac {i e^{3} \pi \,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 ) \mathrm {csgn}\left (i c \right )}{8}+\frac {i e^{2} \pi d \,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 {i e^{2} \pi d \,x^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}+\frac {3 i e \pi \,d^{2} 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 {3 i e \pi \,d^{2} x^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{4}+\frac {\left (e x +d \right )^{4} \ln \left (\left (b x +a \right )^{p}\right )}{4 e}-\frac {i e^{2} \pi d \,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 ) \mathrm {csgn}\left (i c \right )}{2}+\frac {i \pi \,d^{3} x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}+\frac {e^{3} a^{3} p x}{4 b^{3}}-\frac {e^{3} \ln \left (b x +a \right ) a^{4} p}{4 b^{4}}-\frac {3 d^{2} e p \,x^{2}}{4}-\frac {d \,e^{2} p \,x^{3}}{3}+\frac {e^{2} a d p \,x^{2}}{2 b}-\frac {3 e \ln \left (b x +a \right ) a^{2} d^{2} p}{2 b^{2}}\) | \(766\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 207, normalized size = 1.48 \begin {gather*} -\frac {1}{48} \, b p {\left (\frac {3 \, b^{3} x^{4} e^{3} + 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 (x^{4} e^{3} + 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e + 4 \, d^{3} x\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs.
\(2 (131) = 262\).
time = 0.36, size = 266, normalized size = 1.90 \begin {gather*} -\frac {48 \, b^{4} d^{3} p x + {\left (3 \, b^{4} p x^{4} - 4 \, a b^{3} p x^{3} + 6 \, a^{2} b^{2} p x^{2} - 12 \, a^{3} b p x\right )} e^{3} + 8 \, {\left (2 \, b^{4} d p x^{3} - 3 \, a b^{3} d p x^{2} + 6 \, a^{2} b^{2} d p x\right )} e^{2} + 36 \, {\left (b^{4} d^{2} p x^{2} - 2 \, a b^{3} d^{2} p x\right )} e - 12 \, {\left (4 \, b^{4} d^{3} p x + 4 \, a b^{3} d^{3} p + {\left (b^{4} p x^{4} - a^{4} p\right )} e^{3} + 4 \, {\left (b^{4} d p x^{3} + a^{3} b d p\right )} e^{2} + 6 \, {\left (b^{4} d^{2} p x^{2} - a^{2} b^{2} d^{2} p\right )} e\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} x^{4} e^{3} + 4 \, b^{4} d x^{3} e^{2} + 6 \, b^{4} d^{2} x^{2} e + 4 \, b^{4} d^{3} x\right )} \log \left (c\right )}{48 \, b^{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. 335 vs.
\(2 (117) = 234\).
time = 0.95, size = 335, normalized size = 2.39 \begin {gather*} \begin {cases} - \frac {a^{4} e^{3} \log {\left (c \left (a + b x\right )^{p} \right )}}{4 b^{4}} + \frac {a^{3} d e^{2} \log {\left (c \left (a + b x\right )^{p} \right )}}{b^{3}} + \frac {a^{3} e^{3} p x}{4 b^{3}} - \frac {3 a^{2} d^{2} e \log {\left (c \left (a + b x\right )^{p} \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} \log {\left (c \left (a + b x\right )^{p} \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 + d^{3} x \log {\left (c \left (a + 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 (a + b x\right )^{p} \right )}}{2} - \frac {d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log {\left (c \left (a + b x\right )^{p} \right )} - \frac {e^{3} p x^{4}}{16} + \frac {e^{3} x^{4} \log {\left (c \left (a + b x\right )^{p} \right )}}{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} \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. 558 vs.
\(2 (131) = 262\).
time = 3.19, size = 558, normalized size = 3.99 \begin {gather*} \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 \left (c\right )}{b} + \frac {3 \, {\left (b x + a\right )}^{2} d^{2} e \log \left (c\right )}{2 \, b^{2}} - \frac {3 \, {\left (b x + a\right )} a d^{2} e \log \left (c\right )}{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 \left (c\right )}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a d e^{2} \log \left (c\right )}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} d e^{2} \log \left (c\right )}{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 \left (c\right )}{4 \, b^{4}} - \frac {{\left (b x + a\right )}^{3} a e^{3} \log \left (c\right )}{b^{4}} + \frac {3 \, {\left (b x + a\right )}^{2} a^{2} e^{3} \log \left (c\right )}{2 \, b^{4}} - \frac {{\left (b x + a\right )} a^{3} e^{3} \log \left (c\right )}{b^{4}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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