Optimal. Leaf size=583 \[ -\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g \sqrt {f+g x} (e f-d g)^3}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (f+g x)^{3/2} (e f-d g)^2}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (f+g x)^{5/2} (e f-d g)}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {8 b^2 e^{7/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{7 g (e f-d g)^{7/2}}-\frac {128 b^2 e^3 n^2}{105 g \sqrt {f+g x} (e f-d g)^3}-\frac {16 b^2 e^2 n^2}{105 g (f+g x)^{3/2} (e f-d g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.85, antiderivative size = 583, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {2398, 2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 51} \[ -\frac {8 b^2 e^{7/2} n^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g \sqrt {f+g x} (e f-d g)^3}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (f+g x)^{3/2} (e f-d g)^2}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (f+g x)^{5/2} (e f-d g)}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {128 b^2 e^3 n^2}{105 g \sqrt {f+g x} (e f-d g)^3}-\frac {16 b^2 e^2 n^2}{105 g (f+g x)^{3/2} (e f-d g)^2}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{7 g (e f-d g)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 51
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2347
Rule 2348
Rule 2398
Rule 2402
Rule 2411
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac {(4 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{7/2}} \, dx}{7 g}\\ &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac {(4 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{7/2}} \, dx,x,d+e x\right )}{7 g}\\ &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{7/2}} \, dx,x,d+e x\right )}{7 (e f-d g)}+\frac {(4 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{7 g (e f-d g)}\\ &=\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {(4 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{7 (e f-d g)^2}+\frac {\left (4 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{7 g (e f-d g)^2}-\frac {\left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{35 g (e f-d g)}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {\left (4 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{7 (e f-d g)^3}+\frac {\left (4 b e^3 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}-\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{35 g (e f-d g)^2}-\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{21 g (e f-d g)^2}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{35 g (e f-d g)^3}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{21 g (e f-d g)^3}-\frac {\left (4 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac {\left (8 b^2 e^{7/2} n^2\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{7 g (e f-d g)^{7/2}}-\frac {\left (16 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{35 g^2 (e f-d g)^3}-\frac {\left (16 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{21 g^2 (e f-d g)^3}-\frac {\left (16 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{7 g^2 (e f-d g)^3}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac {\left (16 b^2 e^{9/2} n^2\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac {\left (16 b^2 e^{9/2} n^2\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {\left (16 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{7 g (e f-d g)^4}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac {\left (16 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{7 g (e f-d g)^4}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}-\frac {\left (16 b^2 e^{7/2} n^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}-\frac {8 b^2 e^{7/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 3.95, size = 728, normalized size = 1.25 \[ \frac {2 \left (\frac {b e n (f+g x) \left (30 e^{5/2} (f+g x)^{5/2} \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-30 e^{5/2} (f+g x)^{5/2} \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+60 e^2 (f+g x)^2 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )+12 (e f-d g)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+20 e (f+g x) (e f-d g)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )-15 b e^{5/2} n (f+g x)^{5/2} \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )+\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )\right )\right )+15 b e^{5/2} n (f+g x)^{5/2} \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}+1\right )\right )+\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )\right )+120 b e^{5/2} n (f+g x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-40 b e^2 n (f+g x)^2 \sqrt {e f-d g} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )-8 b e n (f+g x) (e f-d g)^{3/2} \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )\right )}{(e f-d g)^{7/2}}-15 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )}{105 g (f+g x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {g x + f} b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, \sqrt {g x + f} a b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt {g x + f} a^{2}}{g^{5} x^{5} + 5 \, f g^{4} x^{4} + 10 \, f^{2} g^{3} x^{3} + 10 \, f^{3} g^{2} x^{2} + 5 \, f^{4} g x + f^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g x +f \right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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