Integrand size = 33, antiderivative size = 975 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{256 c^{9/2} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e} \]
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Time = 25.24 (sec) , antiderivative size = 975, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3781, 6857, 654, 626, 635, 212, 756, 793, 1035, 1092, 1050, 1044, 214, 211} \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2} \tan ^2(d+e x)}{5 c e}+\frac {\left (35 b^2-42 c \tan (d+e x) b-32 a c\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{240 c^3 e}-\frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{3 c e}+\frac {\sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \arctan \left (\frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{256 c^{9/2} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{16 c^{5/2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 \sqrt {c} e}-\frac {\sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \text {arctanh}\left (\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e} \]
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Rule 211
Rule 212
Rule 214
Rule 626
Rule 635
Rule 654
Rule 756
Rule 793
Rule 1035
Rule 1044
Rule 1050
Rule 1092
Rule 3781
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5 \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (-x \sqrt {a+b x+c x^2}+x^3 \sqrt {a+b x+c x^2}+\frac {x \sqrt {a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int x \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int x^3 \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}-\frac {\text {Subst}\left (\int \frac {\frac {b}{2}-(a-c) x-\frac {b x^2}{2}}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int x \left (-2 a-\frac {7 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{5 c e}+\frac {b \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{2 c e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}-\frac {\text {Subst}\left (\int \frac {b+(-a+c) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 e}-\frac {\left (b \left (7 b^2-12 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{32 c^3 e}-\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{16 c^2 e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}+\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^2 e}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{256 c^4 e}+\frac {\text {Subst}\left (\int \frac {-b \sqrt {a^2+b^2-2 a c+c^2}+\left (-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\text {Subst}\left (\int \frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e} \\ & = \frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^4 e}-\frac {\left (b \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac {\left (b \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e} \\ & = \frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{256 c^{9/2} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac {\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.28 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.64 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-\frac {1}{2} \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac {1}{2} \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {c}}+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c}+\frac {b \left (-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}\right )}{2 c}+\frac {\frac {\left (\frac {35 b^2}{4}-8 a c-\frac {21}{2} b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{12 c^2}+\frac {\left (-\frac {35 b^3}{4}+15 a b c\right ) \left (-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}\right )}{8 c^2}}{5 c}}{e} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 3.57 (sec) , antiderivative size = 17768518, normalized size of antiderivative = 18224.12
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 2511 vs. \(2 (876) = 1752\).
Time = 1.25 (sec) , antiderivative size = 5023, normalized size of antiderivative = 5.15 \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \]
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\[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{5}{\left (d + e x \right )}\, dx \]
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\[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{5} \,d x } \]
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Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Hanged} \]
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