Integrand size = 32, antiderivative size = 579 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {8 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{2 d f^5}+\frac {8 b^2 i^3 (f h-e i) (e+f x)^3}{27 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}-\frac {4 b^2 i (f h-e i)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5} \]
[Out]
Time = 1.02 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341, 2356, 45, 2372, 14, 2338} \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {8 b i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))}{9 d f^5}+\frac {i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}-\frac {3 b i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^5}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))^2}{d f^5}-\frac {4 b i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))}{d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}+\frac {(h+i x)^3 (f h-e i) (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {4 a b i x (f h-e i)^3}{d f^4}-\frac {4 b^2 i (e+f x) (f h-e i)^3 \log (c (e+f x))}{d f^5}+\frac {8 b^2 i^3 (e+f x)^3 (f h-e i)}{27 d f^5}+\frac {3 b^2 i^2 (e+f x)^2 (f h-e i)^2}{2 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {8 b^2 i x (f h-e i)^3}{d f^4} \]
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2339
Rule 2341
Rule 2342
Rule 2356
Rule 2367
Rule 2372
Rule 2388
Rule 2458
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^4 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^4 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f} \\ & = \frac {i \text {Subst}\left (\int \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}+\frac {(f h-e i) \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2} \\ & = \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^4 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{2 d f}+\frac {(i (f h-e i)) \text {Subst}\left (\int \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac {(f h-e i)^2 \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3} \\ & = -\frac {2 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}-\frac {2 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {b^2 \text {Subst}\left (\int \frac {48 i (f h-e i)^3+36 i^2 (f h-e i)^2 x+16 i^3 (f h-e i) x^2+3 i^4 x^3+\frac {12 (f h-e i)^4 \log (x)}{x}}{12 f^4} \, dx,x,e+f x\right )}{2 d f}-\frac {(2 b (f h-e i)) \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{3 d f^2}+\frac {\left (i (f h-e i)^2\right ) \text {Subst}\left (\int \left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}+\frac {(f h-e i)^3 \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^4} \\ & = -\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {b^2 \text {Subst}\left (\int \left (48 i (f h-e i)^3+36 i^2 (f h-e i)^2 x+16 i^3 (f h-e i) x^2+3 i^4 x^3+\frac {12 (f h-e i)^4 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{24 d f^5}+\frac {\left (2 b^2 (f h-e i)\right ) \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{6 f^3 x} \, dx,x,e+f x\right )}{3 d f^2}+\frac {\left (i (f h-e i)^2\right ) \text {Subst}\left (\int \left (\frac {(f h-e i) (a+b \log (c x))^2}{f}+\frac {i x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^4}+\frac {\left (i (f h-e i)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {(f h-e i)^4 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^5} \\ & = \frac {2 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{4 d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {\left (b^2 (f h-e i)\right ) \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{x} \, dx,x,e+f x\right )}{9 d f^5}+\frac {\left (i^2 (f h-e i)^2\right ) \text {Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {\left (i (f h-e i)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}-\frac {\left (2 b i (f h-e i)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}+\frac {(f h-e i)^4 \text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^5}+\frac {\left (b^2 (f h-e i)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{2 d f^5} \\ & = -\frac {2 a b i (f h-e i)^3 x}{d f^4}+\frac {2 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{4 d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {b^2 (f h-e i)^4 \log ^2(e+f x)}{4 d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}+\frac {\left (b^2 (f h-e i)\right ) \text {Subst}\left (\int \left (i \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right )+\frac {6 (f h-e i)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{9 d f^5}-\frac {\left (b i^2 (f h-e i)^2\right ) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}-\frac {\left (2 b i (f h-e i)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}-\frac {\left (2 b^2 i (f h-e i)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5} \\ & = -\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {4 b^2 i (f h-e i)^3 x}{d f^4}+\frac {b^2 i^2 (f h-e i)^2 (e+f x)^2}{d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {b^2 (f h-e i)^4 \log ^2(e+f x)}{4 d f^5}-\frac {2 b^2 i (f h-e i)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}+\frac {\left (b^2 i (f h-e i)\right ) \text {Subst}\left (\int \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right ) \, dx,x,e+f x\right )}{9 d f^5}-\frac {\left (2 b^2 i (f h-e i)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5}+\frac {\left (2 b^2 (f h-e i)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^5} \\ & = -\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {8 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{2 d f^5}+\frac {8 b^2 i^3 (f h-e i) (e+f x)^3}{27 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}-\frac {4 b^2 i (f h-e i)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.65 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {3456 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2+2592 i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+1152 i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))^2+216 i^4 (e+f x)^4 (a+b \log (c (e+f x)))^2+\frac {288 (f h-e i)^4 (a+b \log (c (e+f x)))^3}{b}-6912 b i (f h-e i)^3 ((a-b) f x+b (e+f x) \log (c (e+f x)))+1296 b i^2 (f h-e i)^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )+256 b i^3 (f h-e i) \left (b f x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 (a+b \log (c (e+f x)))\right )+27 b i^4 \left (b f x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{864 d f^5} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1134\) vs. \(2(557)=1114\).
Time = 0.92 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.96
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1135\) |
risch | \(\text {Expression too large to display}\) | \(1475\) |
parts | \(\text {Expression too large to display}\) | \(1541\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1891\) |
default | \(\text {Expression too large to display}\) | \(1891\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1912\) |
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Time = 0.30 (sec) , antiderivative size = 939, normalized size of antiderivative = 1.62 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {27 \, {\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} f^{4} i^{4} x^{4} + 4 \, {\left (32 \, {\left (9 \, a^{2} - 6 \, a b + 2 \, b^{2}\right )} f^{4} h i^{3} - {\left (72 \, a^{2} - 84 \, a b + 37 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 288 \, {\left (b^{2} f^{4} h^{4} - 4 \, b^{2} e f^{3} h^{3} i + 6 \, b^{2} e^{2} f^{2} h^{2} i^{2} - 4 \, b^{2} e^{3} f h i^{3} + b^{2} e^{4} i^{4}\right )} \log \left (c f x + c e\right )^{3} + 6 \, {\left (216 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{4} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 30 \, a b + 19 \, b^{2}\right )} e f^{3} h i^{3} + {\left (72 \, a^{2} - 156 \, a b + 115 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 72 \, {\left (3 \, b^{2} f^{4} i^{4} x^{4} + 12 \, a b f^{4} h^{4} - 48 \, {\left (a b - b^{2}\right )} e f^{3} h^{3} i + 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{3} f h i^{3} + {\left (12 \, a b - 25 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (4 \, b^{2} f^{4} h i^{3} - b^{2} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{2} f^{4} h^{2} i^{2} - 4 \, b^{2} e f^{3} h i^{3} + b^{2} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{2} f^{4} h^{3} i - 6 \, b^{2} e f^{3} h^{2} i^{2} + 4 \, b^{2} e^{2} f^{2} h i^{3} - b^{2} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 12 \, {\left (288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{4} h^{3} i - 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{3} f i^{4}\right )} x + 12 \, {\left (9 \, {\left (4 \, a b - b^{2}\right )} f^{4} i^{4} x^{4} + 72 \, a^{2} f^{4} h^{4} - 288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f^{3} h^{3} i + 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{3} f h i^{3} + {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (16 \, {\left (3 \, a b - b^{2}\right )} f^{4} h i^{3} - {\left (12 \, a b - 7 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (36 \, {\left (2 \, a b - b^{2}\right )} f^{4} h^{2} i^{2} - 8 \, {\left (6 \, a b - 5 \, b^{2}\right )} e f^{3} h i^{3} + {\left (12 \, a b - 13 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (48 \, {\left (a b - b^{2}\right )} f^{4} h^{3} i - 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (12 \, a b - 25 \, b^{2}\right )} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )}{864 \, d f^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (534) = 1068\).
Time = 1.88 (sec) , antiderivative size = 1479, normalized size of antiderivative = 2.55 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (557) = 1114\).
Time = 0.27 (sec) , antiderivative size = 1427, normalized size of antiderivative = 2.46 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1213 vs. \(2 (557) = 1114\).
Time = 0.33 (sec) , antiderivative size = 1213, normalized size of antiderivative = 2.09 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
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Time = 2.07 (sec) , antiderivative size = 1346, normalized size of antiderivative = 2.32 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx={\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^4\,x^4}{4\,d\,f^2}-\frac {b^2\,i^3\,x^3\,\left (e\,i-4\,f\,h\right )}{3\,d\,f^3}-\frac {b^2\,i\,x\,\left (e^3\,i^3-4\,e^2\,f\,h\,i^2+6\,e\,f^2\,h^2\,i-4\,f^3\,h^3\right )}{d\,f^5}+\frac {b^2\,i^2\,x^2\,\left (e^2\,i^2-4\,e\,f\,h\,i+6\,f^2\,h^2\right )}{2\,d\,f^4}\right )+\frac {-25\,b^2\,e^4\,i^4+88\,b^2\,e^3\,f\,h\,i^3-108\,b^2\,e^2\,f^2\,h^2\,i^2+48\,b^2\,e\,f^3\,h^3\,i+12\,a\,b\,e^4\,i^4-48\,a\,b\,e^3\,f\,h\,i^3+72\,a\,b\,e^2\,f^2\,h^2\,i^2-48\,a\,b\,e\,f^3\,h^3\,i+12\,a\,b\,f^4\,h^4}{12\,d\,f^5}\right )-x^2\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{2\,f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{24\,d\,f^3}\right )+x^3\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{54\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{24\,d\,f^2}\right )+x\,\left (\frac {288\,a^2\,f^3\,h^3\,i+144\,a\,b\,e^3\,i^4-576\,a\,b\,e^2\,f\,h\,i^3+864\,a\,b\,e\,f^2\,h^2\,i^2-576\,a\,b\,f^3\,h^3\,i-300\,b^2\,e^3\,i^4+1056\,b^2\,e^2\,f\,h\,i^3-1296\,b^2\,e\,f^2\,h^2\,i^2+576\,b^2\,f^3\,h^3\,i}{72\,d\,f^4}+\frac {e\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{12\,d\,f^3}\right )}{f}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x^3\,\left (7\,e\,b^2\,i^4-16\,f\,h\,b^2\,i^3-12\,a\,e\,b\,i^4+48\,a\,f\,h\,b\,i^3\right )}{18\,d\,f^3}-\frac {x^2\,\left (13\,b^2\,e^2\,i^4-40\,b^2\,e\,f\,h\,i^3+36\,b^2\,f^2\,h^2\,i^2-12\,a\,b\,e^2\,i^4+48\,a\,b\,e\,f\,h\,i^3-72\,a\,b\,f^2\,h^2\,i^2\right )}{12\,d\,f^4}+\frac {x\,\left (25\,b^2\,e^3\,i^4-88\,b^2\,e^2\,f\,h\,i^3+108\,b^2\,e\,f^2\,h^2\,i^2-48\,b^2\,f^3\,h^3\,i-12\,a\,b\,e^3\,i^4+48\,a\,b\,e^2\,f\,h\,i^3-72\,a\,b\,e\,f^2\,h^2\,i^2+48\,a\,b\,f^3\,h^3\,i\right )}{6\,d\,f^5}+\frac {b\,i^4\,x^4\,\left (4\,a-b\right )}{8\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (72\,a^2\,e^4\,i^4-288\,a^2\,e^3\,f\,h\,i^3+432\,a^2\,e^2\,f^2\,h^2\,i^2-288\,a^2\,e\,f^3\,h^3\,i+72\,a^2\,f^4\,h^4-300\,a\,b\,e^4\,i^4+1056\,a\,b\,e^3\,f\,h\,i^3-1296\,a\,b\,e^2\,f^2\,h^2\,i^2+576\,a\,b\,e\,f^3\,h^3\,i+415\,b^2\,e^4\,i^4-1360\,b^2\,e^3\,f\,h\,i^3+1512\,b^2\,e^2\,f^2\,h^2\,i^2-576\,b^2\,e\,f^3\,h^3\,i\right )}{72\,d\,f^5}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^4\,i^4-4\,e^3\,f\,h\,i^3+6\,e^2\,f^2\,h^2\,i^2-4\,e\,f^3\,h^3\,i+f^4\,h^4\right )}{3\,d\,f^5}+\frac {i^4\,x^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{32\,d\,f} \]
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