Integrand size = 28, antiderivative size = 229 \[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\frac {e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {2 a}{b p q}} h \sqrt {\frac {\pi }{2}} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}} \]
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Time = 0.47 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2448, 2436, 2337, 2211, 2235, 2437, 2347, 2495} \[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\frac {\sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {\sqrt {\frac {\pi }{2}} h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}} \]
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Rule 2211
Rule 2235
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {g+h x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {h (e+f x)}{f \sqrt {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 ) \\ & = \text {Subst}\left (\frac {h \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {h \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {\left (h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left ((f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {\left (2 h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b p q}+\frac {2 x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (2 (f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {a}{b p q}+\frac {x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {2 a}{b p q}} h \sqrt {\frac {\pi }{2}} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.91 \[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\frac {e^{-\frac {2 a}{b p q}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (2 e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )+\sqrt {2} h (e+f x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )\right )}{2 \sqrt {b} f^2 \sqrt {p} \sqrt {q}} \]
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\[\int \frac {h x +g}{\sqrt {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}}d x\]
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Exception generated. \[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {g + h x}{\sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}}\, dx \]
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\[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {h x + g}{\sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]
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\[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {h x + g}{\sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {g+h\,x}{\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}} \,d x \]
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