Integrand size = 26, antiderivative size = 421 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {4 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {16 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {4 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^3 n^{5/2}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
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Time = 0.98 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2447, 2448, 2436, 2337, 2211, 2235, 2437, 2347} \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {16 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {4 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {4 \sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^3 n^{5/2}}+\frac {8 (d+e x) (f+g x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
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Rule 2211
Rule 2235
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2447
Rule 2448
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {2 \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{b n}-\frac {(4 (e f-d g)) \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b e n} \\ & = -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {12 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 n^2}-\frac {(16 (e f-d g)) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}-\frac {(8 (e f-d g)) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e n^2}+\frac {\left (8 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2} \\ & = -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {12 \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 g (e f-d g) (d+e x)}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^2 (d+e x)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 n^2}-\frac {(16 (e f-d g)) \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{3 b^2 e n^2}-\frac {(8 (e f-d g)) \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 e n^2}+\frac {\left (8 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2} \\ & = -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (12 g^2\right ) \int \frac {(d+e x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac {(16 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac {(8 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac {(24 g (e f-d g)) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac {\left (16 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac {\left (8 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac {\left (12 (e f-d g)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac {\left (8 (e f-d g)^2 (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 )}{3 b^2 e^3 n^3} \\ & = -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (12 g^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {(16 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac {(8 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {(24 g (e f-d g)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {\left (16 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac {\left (8 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {\left (12 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {\left (16 (e f-d g)^2 (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 )}{3 b^3 e^3 n^3} \\ & = \frac {8 e^{-\frac {a}{b n}} (e f-d g)^2 \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 )}{3 b^{5/2} e^3 n^{5/2}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (12 g^2 (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 )}{b^2 e^3 n^3}-\frac {\left (16 g (e f-d g) (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 )}{3 b^2 e^3 n^3}-\frac {\left (8 g (e f-d g) (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 )}{b^2 e^3 n^3}+\frac {\left (24 g (e f-d g) (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 )}{b^2 e^3 n^3}-\frac {\left (16 (e f-d g)^2 (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 )}{3 b^2 e^3 n^3}-\frac {\left (8 (e f-d g)^2 (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 )}{b^2 e^3 n^3}+\frac {\left (12 (e f-d g)^2 (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 )}{b^2 e^3 n^3} \\ & = \frac {8 e^{-\frac {a}{b n}} (e f-d g)^2 \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 )}{3 b^{5/2} e^3 n^{5/2}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (24 g^2 (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^3 e^3 n^3}-\frac {\left (32 g (e f-d g) (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 )}{3 b^3 e^3 n^3}-\frac {\left (16 g (e f-d g) (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^3 e^3 n^3}+\frac {\left (48 g (e f-d g) (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^3 e^3 n^3}-\frac {\left (32 (e f-d g)^2 (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 )}{3 b^3 e^3 n^3}-\frac {\left (16 (e f-d g)^2 (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^3 e^3 n^3}+\frac {\left (24 (e f-d g)^2 (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^3 e^3 n^3} \\ & = \frac {4 e^{-\frac {a}{b n}} (e f-d g)^2 \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 )}{3 b^{5/2} e^3 n^{5/2}}+\frac {16 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (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 )}{3 b^{5/2} e^3 n^{5/2}}+\frac {4 e^{-\frac {3 a}{b n}} g^2 \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 )}{b^{5/2} e^3 n^{5/2}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \\ \end{align*}
Time = 2.64 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=-\frac {2 e^{-\frac {3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (2 d e^{\frac {2 a}{b n}} g (8 e f+d g) \sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+8 e^{\frac {a}{b n}} g (-e f+d g) \sqrt {2 \pi } (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-6 g^2 \sqrt {3 \pi } (d+e x)^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+\sqrt {b} e^{\frac {2 a}{b n}} \sqrt {n} \left (c (d+e x)^n\right )^{2/n} \left (2 b \left (e^2 f^2+6 d e f g+2 d^2 g^2\right ) n \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2}+e e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} (f+g x) \left (b e n (f+g x)+2 a (e f+2 d g+3 e g x)+2 b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )\right )\right )\right )}{3 b^{5/2} e^3 n^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
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\[\int \frac {\left (g x +f \right )^{2}}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]
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