Integrand size = 28, antiderivative size = 103 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=-\frac {4 b p q \sqrt {g+h x}}{h}+\frac {4 b \sqrt {f g-e h} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h} \]
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Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2442, 52, 65, 214, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {4 b p q \sqrt {f g-e h} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}-\frac {4 b p q \sqrt {g+h x}}{h} \]
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Rule 52
Rule 65
Rule 214
Rule 2442
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {\sqrt {g+h x}}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {4 b p q \sqrt {g+h x}}{h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\text {Subst}\left (\frac {(2 b (f g-e h) p q) \int \frac {1}{(e+f x) \sqrt {g+h x}} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {4 b p q \sqrt {g+h x}}{h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\text {Subst}\left (\frac {(4 b (f g-e h) p q) \text {Subst}\left (\int \frac {1}{e-\frac {f g}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {4 b p q \sqrt {g+h x}}{h}+\frac {4 b \sqrt {f g-e h} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f} h}+\frac {2 \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\frac {2 \left (\frac {2 b \sqrt {f g-e h} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f}}+\sqrt {g+h x} \left (a-2 b p q+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{h} \]
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Time = 0.65 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {2 \sqrt {h x +g}\, a +2 b \left (\ln \left (c \left (d \left (\frac {f \left (h x +g \right )+e h -f g}{h}\right )^{p}\right )^{q}\right ) \sqrt {h x +g}-2 q p f \left (\frac {\sqrt {h x +g}}{f}+\frac {\left (-e h +f g \right ) \arctan \left (\frac {f \sqrt {h x +g}}{\sqrt {\left (e h -f g \right ) f}}\right )}{f \sqrt {\left (e h -f g \right ) f}}\right )\right )}{h}\) | \(118\) |
default | \(\frac {2 \sqrt {h x +g}\, a +2 b \left (\ln \left (c \left (d \left (\frac {f \left (h x +g \right )+e h -f g}{h}\right )^{p}\right )^{q}\right ) \sqrt {h x +g}-2 q p f \left (\frac {\sqrt {h x +g}}{f}+\frac {\left (-e h +f g \right ) \arctan \left (\frac {f \sqrt {h x +g}}{\sqrt {\left (e h -f g \right ) f}}\right )}{f \sqrt {\left (e h -f g \right ) f}}\right )\right )}{h}\) | \(118\) |
parts | \(\frac {2 a \sqrt {h x +g}}{h}+\frac {2 b \left (\ln \left (c \left (d \left (\frac {f \left (h x +g \right )+e h -f g}{h}\right )^{p}\right )^{q}\right ) \sqrt {h x +g}-2 q p f \left (\frac {\sqrt {h x +g}}{f}+\frac {\left (-e h +f g \right ) \arctan \left (\frac {f \sqrt {h x +g}}{\sqrt {\left (e h -f g \right ) f}}\right )}{f \sqrt {\left (e h -f g \right ) f}}\right )\right )}{h}\) | \(121\) |
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Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\left [\frac {2 \, {\left (b p q \sqrt {\frac {f g - e h}{f}} \log \left (\frac {f h x + 2 \, f g - e h + 2 \, \sqrt {h x + g} f \sqrt {\frac {f g - e h}{f}}}{f x + e}\right ) + {\left (b p q \log \left (f x + e\right ) - 2 \, b p q + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h}, \frac {2 \, {\left (2 \, b p q \sqrt {-\frac {f g - e h}{f}} \arctan \left (-\frac {\sqrt {h x + g} f \sqrt {-\frac {f g - e h}{f}}}{f g - e h}\right ) + {\left (b p q \log \left (f x + e\right ) - 2 \, b p q + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h}\right ] \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g + h x}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=-\frac {2 \, {\left ({\left (2 \, f {\left (\frac {{\left (f g - e h\right )} \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + e f h}}\right )}{\sqrt {-f^{2} g + e f h} f} + \frac {\sqrt {h x + g}}{f}\right )} - \sqrt {h x + g} \log \left (f x + e\right )\right )} b p q - \sqrt {h x + g} b q \log \left (d\right ) - \sqrt {h x + g} b \log \left (c\right ) - \sqrt {h x + g} a\right )}}{h} \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {g+h\,x}} \,d x \]
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