Optimal. Leaf size=975 \[ \frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2} \tan ^2(d+e x)}{5 c e}+\frac {\left (35 b^2-42 c \tan (d+e x) b-32 a c\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{240 c^3 e}-\frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{3 c e}+\frac {\sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{256 c^{9/2} e}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{16 c^{5/2} e}+\frac {b \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 \sqrt {c} e}-\frac {\sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e} \]
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Rubi [A] time = 24.46, antiderivative size = 975, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3700, 6725, 640, 612, 621, 206, 742, 779, 1021, 1078, 1036, 1030, 208, 205} \[ \frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2} \tan ^2(d+e x)}{5 c e}+\frac {\left (35 b^2-42 c \tan (d+e x) b-32 a c\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{240 c^3 e}-\frac {\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{3 c e}+\frac {\sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac {b^2-\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{256 c^{9/2} e}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{16 c^{5/2} e}+\frac {b \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 \sqrt {c} e}-\frac {\sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac {b^2+\sqrt {a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac {b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac {b (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac {\sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 208
Rule 612
Rule 621
Rule 640
Rule 742
Rule 779
Rule 1021
Rule 1030
Rule 1036
Rule 1078
Rule 3700
Rule 6725
Rubi steps
\begin {align*} \int \tan ^5(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int x \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\operatorname {Subst}\left (\int x^3 \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 \operatorname {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 {\operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{256 c^4 e}+\frac {\operatorname {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 {\operatorname {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 \tanh ^{-1}\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 ) \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 )} \tan ^{-1}\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 \tanh ^{-1}\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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 )} \tanh ^{-1}\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*}
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Mathematica [C] time = 6.25, size = 690, normalized size = 0.71 \[ \frac {\frac {b \left (\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}\right )}{2 c}+\frac {\frac {\left (-8 a c+\frac {35 b^2}{4}-\frac {21}{2} b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{12 c^2}+\frac {\left (15 a b c-\frac {35 b^3}{4}\right ) \left (\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}\right )}{8 c^2}}{5 c}+\frac {\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c}-\frac {\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c}+\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}+\frac {1}{4} \left (\frac {(b-2 i c) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \sqrt {a-i b-c} \tanh ^{-1}\left (\frac {2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )+\frac {1}{4} \left (\frac {(b+2 i c) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \sqrt {a+i b-c} \tanh ^{-1}\left (\frac {2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )}{e} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [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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maple [B] time = 2.13, size = 17767396, normalized size = 18222.97 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{5}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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