Integrand size = 26, antiderivative size = 383 \[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}}+\frac {3 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}} \]
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Time = 0.51 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2448, 2436, 2337, 2211, 2235, 2437, 2347} \[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {\sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {3 \sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}} \]
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Rule 2211
Rule 2235
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(e f-d g)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g (e f-d g)^2 (d+e x)}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g^2 (e f-d g) (d+e x)^2}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^3 (d+e x)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx \\ & = \frac {g^3 \int \frac {(d+e x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {\left (3 g^2 (e f-d g)\right ) \int \frac {(d+e x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {\left (3 g (e f-d g)^2\right ) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3} \\ & = \frac {g^3 \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g^2 (e f-d g)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {(e f-d g)^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4} \\ & = \frac {\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left ((e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n} \\ & = \frac {\left (2 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b n}+\frac {4 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (6 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b n}+\frac {3 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (6 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b n}+\frac {2 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (2 (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b e^4 n} \\ & = \frac {e^{-\frac {a}{b n}} (e f-d g)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}}+\frac {3 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.86 \[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {4 a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (2 e^{\frac {3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+g^3 (d+e x)^3 \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+3 \sqrt {2} e^{\frac {2 a}{b n}} g (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {3} e^{\frac {a}{b n}} g^2 (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{2 \sqrt {b} e^4 \sqrt {n}} \]
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\[\int \frac {\left (g x +f \right )^{3}}{\sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}d x\]
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Exception generated. \[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {\left (f + g x\right )^{3}}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \]
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\[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {{\left (g x + f\right )}^{3}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
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\[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {{\left (g x + f\right )}^{3}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {{\left (f+g\,x\right )}^3}{\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]
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