Optimal. Leaf size=510 \[ \frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {16 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 e^{3/2} g}+\frac {8 b^2 (e f-d g)^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 e^{3/2} g} \]
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Rubi [A]
time = 1.05, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used =
{2445, 2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52}
\begin {gather*} \frac {8 b^2 n^2 (e f-d g)^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 e^{3/2} g}+\frac {8 b n (e f-d g)^{3/2} \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 )}{3 e^{3/2} g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}-\frac {8 b n \sqrt {f+g x} (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {8 b^2 n^2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {64 b^2 n^2 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}+\frac {16 b^2 n^2 (e f-d g)^{3/2} \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 )}{3 e^{3/2} g}+\frac {64 b^2 n^2 \sqrt {f+g x} (e f-d g)}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2388
Rule 2390
Rule 2445
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps
\begin {align*} \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {(4 b e n) \int \frac {(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{3 g}\\ &=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {(4 b n) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{3 g}\\ &=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {(4 b n) \text {Subst}\left (\int \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{3 e}-\frac {(4 b (e f-d g) n) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{3 e g}\\ &=-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {(4 b (e f-d g) n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 e^2}-\frac {\left (4 b (e f-d g)^2 n\right ) \text {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 )}{3 e^2 g}+\frac {\left (8 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}}{x} \, dx,x,d+e x\right )}{9 g}\\ &=\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {\left (8 b^2 (e f-d g) n^2\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{9 e g}+\frac {\left (8 b^2 (e f-d g) n^2\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{3 e g}+\frac {\left (4 b^2 (e f-d g)^2 n^2\right ) \text {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 )}{3 e^2 g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {\left (8 b^2 (e f-d g)^{3/2} n^2\right ) \text {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 )}{3 e^{3/2} g}+\frac {\left (8 b^2 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{9 e^2 g}+\frac {\left (8 b^2 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 e^2 g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {\left (16 b^2 (e f-d g)^{3/2} n^2\right ) \text {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 )}{3 \sqrt {e} g}+\frac {\left (16 b^2 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{9 e g^2}+\frac {\left (16 b^2 (e f-d g)^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 e g^2}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {\left (16 b^2 (e f-d g)^{3/2} n^2\right ) \text {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 )}{3 \sqrt {e} g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {\left (16 b^2 (e f-d g) n^2\right ) \text {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 )}{3 e g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {16 b^2 (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}-\frac {\left (16 b^2 (e f-d g) n^2\right ) \text {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 )}{3 e g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {16 b^2 (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {\left (16 b^2 (e f-d g)^{3/2} n^2\right ) \text {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 )}{3 e^{3/2} g}\\ &=\frac {64 b^2 (e f-d g) n^2 \sqrt {f+g x}}{9 e g}+\frac {16 b^2 n^2 (f+g x)^{3/2}}{27 g}-\frac {64 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{9 e^{3/2} g}-\frac {8 b^2 (e f-d g)^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 e^{3/2} g}-\frac {8 b (e f-d g) n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e g}-\frac {8 b n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {8 b (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {16 b^2 (e f-d g)^{3/2} 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 )}{3 e^{3/2} g}+\frac {8 b^2 (e f-d g)^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 e^{3/2} g}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.11, size = 351, normalized size = 0.69 \begin {gather*} \frac {2 \left (\frac {3 b^2 n^2 \sqrt {f+g x} \left (3 g (d+e x) \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;\frac {g (d+e x)}{-e f+d g}\right )+\log (d+e x) \left (-3 g (d+e x) \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {g (d+e x)}{-e f+d g}\right )+\left (d g+e g x \sqrt {\frac {e (f+g x)}{e f-d g}}+e f \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right ) \log (d+e x)\right )\right )}{e \sqrt {\frac {e (f+g x)}{e f-d g}}}-\frac {2 b n \left (6 (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+\sqrt {e} \sqrt {f+g x} (6 d g-2 e (4 f+g x)+3 e (f+g x) \log (d+e x))\right ) \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}+3 (f+g x)^{3/2} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2\right )}{9 g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \sqrt {g x +f}\, \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \sqrt {f + g x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {f+g\,x}\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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