Integrand size = 26, antiderivative size = 158 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=-\frac {b (f g-e h)^3 p q x}{4 f^3}-\frac {b (f g-e h)^2 p q (g+h x)^2}{8 f^2 h}-\frac {b (f g-e h) p q (g+h x)^3}{12 f h}-\frac {b p q (g+h x)^4}{16 h}-\frac {b (f g-e h)^4 p q \log (e+f x)}{4 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 h} \]
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Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2442, 45, 2495} \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 h}-\frac {b p q (f g-e h)^4 \log (e+f x)}{4 f^4 h}-\frac {b p q x (f g-e h)^3}{4 f^3}-\frac {b p q (g+h x)^2 (f g-e h)^2}{8 f^2 h}-\frac {b p q (g+h x)^3 (f g-e h)}{12 f h}-\frac {b p q (g+h x)^4}{16 h} \]
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Rule 45
Rule 2442
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (g+h x)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 h}-\text {Subst}\left (\frac {(b f p q) \int \frac {(g+h x)^4}{e+f x} \, dx}{4 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 h}-\text {Subst}\left (\frac {(b f p q) \int \left (\frac {h (f g-e h)^3}{f^4}+\frac {(f g-e h)^4}{f^4 (e+f x)}+\frac {h (f g-e h)^2 (g+h x)}{f^3}+\frac {h (f g-e h) (g+h x)^2}{f^2}+\frac {h (g+h x)^3}{f}\right ) \, dx}{4 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {b (f g-e h)^3 p q x}{4 f^3}-\frac {b (f g-e h)^2 p q (g+h x)^2}{8 f^2 h}-\frac {b (f g-e h) p q (g+h x)^3}{12 f h}-\frac {b p q (g+h x)^4}{16 h}-\frac {b (f g-e h)^4 p q \log (e+f x)}{4 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 h} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.47 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {f x \left (12 a f^3 \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right )-b p q \left (-12 e^3 h^3+6 e^2 f h^2 (8 g+h x)-4 e f^2 h \left (18 g^2+6 g h x+h^2 x^2\right )+f^3 \left (48 g^3+36 g^2 h x+16 g h^2 x^2+3 h^3 x^3\right )\right )\right )-12 b e^2 h \left (6 f^2 g^2-4 e f g h+e^2 h^2\right ) p q \log (e+f x)+12 b f^3 \left (4 e g^3+f x \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right )\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )}{48 f^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs. \(2(146)=292\).
Time = 5.98 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.95
method | result | size |
parallelrisch | \(-\frac {-48 \ln \left (f x +e \right ) b \,e^{3} f g \,h^{2} p q +72 \ln \left (f x +e \right ) b \,e^{2} f^{2} g^{2} h p q -24 x^{2} b e \,f^{3} g \,h^{2} p q +48 x b \,e^{2} f^{2} g \,h^{2} p q -48 x^{3} a \,f^{4} g \,h^{2}-72 x^{2} a \,f^{4} g^{2} h -48 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b \,f^{4} g^{3}+48 \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b e \,f^{3} g^{3}-48 b e \,f^{3} g^{3} p q +12 b \,e^{4} h^{3} p q -72 x b e \,f^{3} g^{2} h p q +72 b \,e^{2} g^{2} h p q \,f^{2}-48 b \,e^{3} f g \,h^{2} p q -12 x^{4} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b \,f^{4} h^{3}+48 a e \,g^{3} f^{3}-72 x^{2} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b \,f^{4} g^{2} h +48 x b \,f^{4} g^{3} p q +12 \ln \left (f x +e \right ) b \,e^{4} h^{3} p q +3 x^{4} b \,f^{4} h^{3} p q -48 x^{3} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b \,f^{4} g \,h^{2}-12 x^{4} a \,f^{4} h^{3}-48 x a \,f^{4} g^{3}+36 x^{2} b \,f^{4} g^{2} h p q -12 x b \,e^{3} f \,h^{3} p q -96 \ln \left (f x +e \right ) b e \,f^{3} g^{3} p q -4 x^{3} b e \,f^{3} h^{3} p q +16 x^{3} b \,f^{4} g \,h^{2} p q +6 x^{2} b \,e^{2} f^{2} h^{3} p q}{48 f^{4}}\) | \(466\) |
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (146) = 292\).
Time = 0.36 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.56 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=-\frac {3 \, {\left (b f^{4} h^{3} p q - 4 \, a f^{4} h^{3}\right )} x^{4} - 4 \, {\left (12 \, a f^{4} g h^{2} - {\left (4 \, b f^{4} g h^{2} - b e f^{3} h^{3}\right )} p q\right )} x^{3} - 6 \, {\left (12 \, a f^{4} g^{2} h - {\left (6 \, b f^{4} g^{2} h - 4 \, b e f^{3} g h^{2} + b e^{2} f^{2} h^{3}\right )} p q\right )} x^{2} - 12 \, {\left (4 \, a f^{4} g^{3} - {\left (4 \, b f^{4} g^{3} - 6 \, b e f^{3} g^{2} h + 4 \, b e^{2} f^{2} g h^{2} - b e^{3} f h^{3}\right )} p q\right )} x - 12 \, {\left (b f^{4} h^{3} p q x^{4} + 4 \, b f^{4} g h^{2} p q x^{3} + 6 \, b f^{4} g^{2} h p q x^{2} + 4 \, b f^{4} g^{3} p q x + {\left (4 \, b e f^{3} g^{3} - 6 \, b e^{2} f^{2} g^{2} h + 4 \, b e^{3} f g h^{2} - b e^{4} h^{3}\right )} p q\right )} \log \left (f x + e\right ) - 12 \, {\left (b f^{4} h^{3} x^{4} + 4 \, b f^{4} g h^{2} x^{3} + 6 \, b f^{4} g^{2} h x^{2} + 4 \, b f^{4} g^{3} x\right )} \log \left (c\right ) - 12 \, {\left (b f^{4} h^{3} q x^{4} + 4 \, b f^{4} g h^{2} q x^{3} + 6 \, b f^{4} g^{2} h q x^{2} + 4 \, b f^{4} g^{3} q x\right )} \log \left (d\right )}{48 \, f^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (139) = 278\).
Time = 2.50 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.89 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\begin {cases} a g^{3} x + \frac {3 a g^{2} h x^{2}}{2} + a g h^{2} x^{3} + \frac {a h^{3} x^{4}}{4} - \frac {b e^{4} h^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{4 f^{4}} + \frac {b e^{3} g h^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{3}} + \frac {b e^{3} h^{3} p q x}{4 f^{3}} - \frac {3 b e^{2} g^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{2}} - \frac {b e^{2} g h^{2} p q x}{f^{2}} - \frac {b e^{2} h^{3} p q x^{2}}{8 f^{2}} + \frac {b e g^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {3 b e g^{2} h p q x}{2 f} + \frac {b e g h^{2} p q x^{2}}{2 f} + \frac {b e h^{3} p q x^{3}}{12 f} - b g^{3} p q x + b g^{3} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {3 b g^{2} h p q x^{2}}{4} + \frac {3 b g^{2} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} - \frac {b g h^{2} p q x^{3}}{3} + b g h^{2} x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {b h^{3} p q x^{4}}{16} + \frac {b h^{3} x^{4} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{4} & \text {for}\: f \neq 0 \\\left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right ) \left (g^{3} x + \frac {3 g^{2} h x^{2}}{2} + g h^{2} x^{3} + \frac {h^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (146) = 292\).
Time = 0.20 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.92 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {1}{4} \, b h^{3} x^{4} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {1}{4} \, a h^{3} x^{4} - b f g^{3} p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} - \frac {1}{48} \, b f h^{3} p q {\left (\frac {12 \, e^{4} \log \left (f x + e\right )}{f^{5}} + \frac {3 \, f^{3} x^{4} - 4 \, e f^{2} x^{3} + 6 \, e^{2} f x^{2} - 12 \, e^{3} x}{f^{4}}\right )} + \frac {1}{6} \, b f g h^{2} p q {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{f^{3}}\right )} - \frac {3}{4} \, b f g^{2} h p q {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}} + \frac {f x^{2} - 2 \, e x}{f^{2}}\right )} + b g h^{2} x^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a g h^{2} x^{3} + \frac {3}{2} \, b g^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {3}{2} \, a g^{2} h x^{2} + b g^{3} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a g^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 987 vs. \(2 (146) = 292\).
Time = 0.34 (sec) , antiderivative size = 987, normalized size of antiderivative = 6.25 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\text {Too large to display} \]
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Time = 1.54 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.34 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (b\,g^3\,x+\frac {3\,b\,g^2\,h\,x^2}{2}+b\,g\,h^2\,x^3+\frac {b\,h^3\,x^4}{4}\right )-x^2\,\left (\frac {e\,\left (\frac {h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {e\,h^3\,\left (4\,a-b\,p\,q\right )}{4\,f}\right )}{2\,f}-\frac {3\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{4\,f}\right )+x\,\left (\frac {4\,a\,f\,g^3+12\,a\,e\,g^2\,h-4\,b\,f\,g^3\,p\,q}{4\,f}+\frac {e\,\left (\frac {e\,\left (\frac {h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {e\,h^3\,\left (4\,a-b\,p\,q\right )}{4\,f}\right )}{f}-\frac {3\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{2\,f}\right )}{f}\right )+x^3\,\left (\frac {h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{3\,f}-\frac {e\,h^3\,\left (4\,a-b\,p\,q\right )}{12\,f}\right )-\frac {\ln \left (e+f\,x\right )\,\left (b\,p\,q\,e^4\,h^3-4\,b\,p\,q\,e^3\,f\,g\,h^2+6\,b\,p\,q\,e^2\,f^2\,g^2\,h-4\,b\,p\,q\,e\,f^3\,g^3\right )}{4\,f^4}+\frac {h^3\,x^4\,\left (4\,a-b\,p\,q\right )}{16} \]
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