Optimal. Leaf size=502 \[ \frac{a b d^2 x}{2 c^3}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}+\frac{a b e^2 x}{4 c^7}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d e x^4}{30 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{23 b^2 d e \log \left (c^2 x^2+1\right )}{45 c^6}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x^6}{168 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{71 b^2 e^2 x^2}{840 c^6}-\frac{22 b^2 e^2 \log \left (c^2 x^2+1\right )}{105 c^8}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7} \]
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Rubi [A] time = 1.14011, antiderivative size = 502, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4980, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac{a b d^2 x}{2 c^3}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}+\frac{a b e^2 x}{4 c^7}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d e x^4}{30 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{23 b^2 d e \log \left (c^2 x^2+1\right )}{45 c^6}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x^6}{168 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{71 b^2 e^2 x^2}{840 c^6}-\frac{22 b^2 e^2 \log \left (c^2 x^2+1\right )}{105 c^8}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 4916
Rule 266
Rule 43
Rule 4846
Rule 260
Rule 4884
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^7 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{3} (2 b c d e) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{4} \left (b c e^2\right ) \int \frac{x^8 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}-\frac{(2 b d e) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{(2 b d e) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}-\frac{\left (b e^2\right ) \int x^6 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c}+\frac{\left (b e^2\right ) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} \left (b^2 d^2\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac{1}{15} \left (2 b^2 d e\right ) \int \frac{x^5}{1+c^2 x^2} \, dx+\frac{(2 b d e) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{(2 b d e) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}+\frac{1}{28} \left (b^2 e^2\right ) \int \frac{x^7}{1+c^2 x^2} \, dx+\frac{\left (b e^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^3}-\frac{\left (b e^2\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c^3}\\ &=\frac{a b d^2 x}{2 c^3}-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}+\frac{1}{15} \left (b^2 d e\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac{(2 b d e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac{(2 b d e) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac{\left (2 b^2 d e\right ) \int \frac{x^3}{1+c^2 x^2} \, dx}{9 c^2}+\frac{1}{56} \left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+c^2 x} \, dx,x,x^2\right )-\frac{\left (b e^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^5}+\frac{\left (b e^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c^5}-\frac{\left (b^2 e^2\right ) \int \frac{x^5}{1+c^2 x^2} \, dx}{20 c^2}\\ &=\frac{a b d^2 x}{2 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 d^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{2 c^2}+\frac{1}{15} \left (b^2 d e\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (2 b^2 d e\right ) \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac{\left (b^2 d e\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{9 c^2}+\frac{1}{56} \left (b^2 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^6}-\frac{x}{c^4}+\frac{x^2}{c^2}-\frac{1}{c^6 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{\left (b e^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^7}-\frac{\left (b e^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{4 c^7}+\frac{\left (b^2 e^2\right ) \int \frac{x^3}{1+c^2 x^2} \, dx}{12 c^4}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )}{40 c^2}\\ &=\frac{a b d^2 x}{2 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{a b e^2 x}{4 c^7}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{b^2 d e x^2}{15 c^4}+\frac{b^2 e^2 x^2}{56 c^6}+\frac{b^2 d e x^4}{30 c^2}-\frac{b^2 e^2 x^4}{112 c^4}+\frac{b^2 e^2 x^6}{168 c^2}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{b^2 d e \log \left (1+c^2 x^2\right )}{15 c^6}-\frac{b^2 e^2 \log \left (1+c^2 x^2\right )}{56 c^8}+\frac{\left (2 b^2 d e\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 c^4}-\frac{\left (b^2 d e\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{9 c^2}+\frac{\left (b^2 e^2\right ) \int \tan ^{-1}(c x) \, dx}{4 c^7}+\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{24 c^4}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{40 c^2}\\ &=\frac{a b d^2 x}{2 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{a b e^2 x}{4 c^7}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{3 b^2 e^2 x^2}{70 c^6}+\frac{b^2 d e x^4}{30 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{b^2 e^2 x^6}{168 c^2}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{23 b^2 d e \log \left (1+c^2 x^2\right )}{45 c^6}-\frac{3 b^2 e^2 \log \left (1+c^2 x^2\right )}{70 c^8}-\frac{\left (b^2 e^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{4 c^6}+\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{24 c^4}\\ &=\frac{a b d^2 x}{2 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{a b e^2 x}{4 c^7}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{71 b^2 e^2 x^2}{840 c^6}+\frac{b^2 d e x^4}{30 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{b^2 e^2 x^6}{168 c^2}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}-\frac{b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac{2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac{b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{23 b^2 d e \log \left (1+c^2 x^2\right )}{45 c^6}-\frac{22 b^2 e^2 \log \left (1+c^2 x^2\right )}{105 c^8}\\ \end{align*}
Mathematica [A] time = 0.455738, size = 414, normalized size = 0.82 \[ \frac{c x \left (105 a^2 c^7 x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-2 a b \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )-7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )+105 c^2 e \left (8 d+e x^2\right )-315 e^2\right )+b^2 c x \left (3 c^4 \left (70 d^2+28 d e x^2+5 e^2 x^4\right )-2 c^2 e \left (224 d+27 e x^2\right )+213 e^2\right )\right )+2 b \tan ^{-1}(c x) \left (105 a \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )+b c x \left (-3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )-105 c^2 e \left (8 d+e x^2\right )+315 e^2\right )\right )-8 b^2 \left (105 c^4 d^2-161 c^2 d e+66 e^2\right ) \log \left (c^2 x^2+1\right )+105 b^2 \tan ^{-1}(c x)^2 \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )}{2520 c^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 621, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71036, size = 697, normalized size = 1.39 \begin{align*} \frac{1}{8} \, b^{2} e^{2} x^{8} \arctan \left (c x\right )^{2} + \frac{1}{8} \, a^{2} e^{2} x^{8} + \frac{1}{3} \, b^{2} d e x^{6} \arctan \left (c x\right )^{2} + \frac{1}{3} \, a^{2} d e x^{6} + \frac{1}{4} \, b^{2} d^{2} x^{4} \arctan \left (c x\right )^{2} + \frac{1}{4} \, a^{2} d^{2} x^{4} + \frac{1}{6} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d^{2} - \frac{1}{12} \,{\left (2 \, c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac{c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d^{2} + \frac{2}{45} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b d e - \frac{1}{90} \,{\left (4 \, c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac{3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} d e + \frac{1}{420} \,{\left (105 \, x^{8} \arctan \left (c x\right ) - c{\left (\frac{15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac{105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} a b e^{2} - \frac{1}{840} \,{\left (2 \, c{\left (\frac{15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac{105 \, \arctan \left (c x\right )}{c^{9}}\right )} \arctan \left (c x\right ) - \frac{5 \, c^{6} x^{6} - 18 \, c^{4} x^{4} + 71 \, c^{2} x^{2} + 105 \, \arctan \left (c x\right )^{2} - 176 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )} b^{2} e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13067, size = 1168, normalized size = 2.33 \begin{align*} \frac{315 \, a^{2} c^{8} e^{2} x^{8} - 90 \, a b c^{7} e^{2} x^{7} + 15 \,{\left (56 \, a^{2} c^{8} d e + b^{2} c^{6} e^{2}\right )} x^{6} - 42 \,{\left (8 \, a b c^{7} d e - 3 \, a b c^{5} e^{2}\right )} x^{5} + 6 \,{\left (105 \, a^{2} c^{8} d^{2} + 14 \, b^{2} c^{6} d e - 9 \, b^{2} c^{4} e^{2}\right )} x^{4} - 70 \,{\left (6 \, a b c^{7} d^{2} - 8 \, a b c^{5} d e + 3 \, a b c^{3} e^{2}\right )} x^{3} +{\left (210 \, b^{2} c^{6} d^{2} - 448 \, b^{2} c^{4} d e + 213 \, b^{2} c^{2} e^{2}\right )} x^{2} + 105 \,{\left (3 \, b^{2} c^{8} e^{2} x^{8} + 8 \, b^{2} c^{8} d e x^{6} + 6 \, b^{2} c^{8} d^{2} x^{4} - 6 \, b^{2} c^{4} d^{2} + 8 \, b^{2} c^{2} d e - 3 \, b^{2} e^{2}\right )} \arctan \left (c x\right )^{2} + 210 \,{\left (6 \, a b c^{5} d^{2} - 8 \, a b c^{3} d e + 3 \, a b c e^{2}\right )} x + 2 \,{\left (315 \, a b c^{8} e^{2} x^{8} + 840 \, a b c^{8} d e x^{6} - 45 \, b^{2} c^{7} e^{2} x^{7} + 630 \, a b c^{8} d^{2} x^{4} - 630 \, a b c^{4} d^{2} + 840 \, a b c^{2} d e - 21 \,{\left (8 \, b^{2} c^{7} d e - 3 \, b^{2} c^{5} e^{2}\right )} x^{5} - 315 \, a b e^{2} - 35 \,{\left (6 \, b^{2} c^{7} d^{2} - 8 \, b^{2} c^{5} d e + 3 \, b^{2} c^{3} e^{2}\right )} x^{3} + 105 \,{\left (6 \, b^{2} c^{5} d^{2} - 8 \, b^{2} c^{3} d e + 3 \, b^{2} c e^{2}\right )} x\right )} \arctan \left (c x\right ) - 8 \,{\left (105 \, b^{2} c^{4} d^{2} - 161 \, b^{2} c^{2} d e + 66 \, b^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{2520 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.1747, size = 758, normalized size = 1.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.53245, size = 879, normalized size = 1.75 \begin{align*} \frac{315 \, b^{2} c^{8} x^{8} \arctan \left (c x\right )^{2} e^{2} + 630 \, a b c^{8} x^{8} \arctan \left (c x\right ) e^{2} + 840 \, b^{2} c^{8} d x^{6} \arctan \left (c x\right )^{2} e + 315 \, a^{2} c^{8} x^{8} e^{2} + 1680 \, a b c^{8} d x^{6} \arctan \left (c x\right ) e + 630 \, b^{2} c^{8} d^{2} x^{4} \arctan \left (c x\right )^{2} - 90 \, b^{2} c^{7} x^{7} \arctan \left (c x\right ) e^{2} + 840 \, a^{2} c^{8} d x^{6} e + 1260 \, a b c^{8} d^{2} x^{4} \arctan \left (c x\right ) - 90 \, a b c^{7} x^{7} e^{2} - 336 \, b^{2} c^{7} d x^{5} \arctan \left (c x\right ) e + 630 \, a^{2} c^{8} d^{2} x^{4} - 336 \, a b c^{7} d x^{5} e - 420 \, b^{2} c^{7} d^{2} x^{3} \arctan \left (c x\right ) + 15 \, b^{2} c^{6} x^{6} e^{2} - 420 \, a b c^{7} d^{2} x^{3} + 126 \, b^{2} c^{5} x^{5} \arctan \left (c x\right ) e^{2} + 84 \, b^{2} c^{6} d x^{4} e + 126 \, a b c^{5} x^{5} e^{2} + 560 \, b^{2} c^{5} d x^{3} \arctan \left (c x\right ) e + 210 \, b^{2} c^{6} d^{2} x^{2} + 560 \, a b c^{5} d x^{3} e + 1260 \, b^{2} c^{5} d^{2} x \arctan \left (c x\right ) - 54 \, b^{2} c^{4} x^{4} e^{2} + 1260 \, a b c^{5} d^{2} x - 630 \, b^{2} c^{4} d^{2} \arctan \left (c x\right )^{2} - 210 \, b^{2} c^{3} x^{3} \arctan \left (c x\right ) e^{2} - 448 \, b^{2} c^{4} d x^{2} e - 1260 \, a b c^{4} d^{2} \arctan \left (c x\right ) - 210 \, a b c^{3} x^{3} e^{2} - 1680 \, b^{2} c^{3} d x \arctan \left (c x\right ) e - 840 \, b^{2} c^{4} d^{2} \log \left (c^{2} x^{2} + 1\right ) - 1680 \, \pi a b c^{2} d e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 1680 \, a b c^{3} d x e + 840 \, b^{2} c^{2} d \arctan \left (c x\right )^{2} e + 213 \, b^{2} c^{2} x^{2} e^{2} + 1680 \, a b c^{2} d \arctan \left (c x\right ) e + 1288 \, b^{2} c^{2} d e \log \left (c^{2} x^{2} + 1\right ) + 630 \, b^{2} c x \arctan \left (c x\right ) e^{2} + 630 \, a b c x e^{2} - 315 \, b^{2} \arctan \left (c x\right )^{2} e^{2} - 630 \, a b \arctan \left (c x\right ) e^{2} - 528 \, b^{2} e^{2} \log \left (c^{2} x^{2} + 1\right )}{2520 \, c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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