Optimal. Leaf size=339 \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}-\frac{i \sqrt{a-i b} (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i \sqrt{a+i b} (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f} \]
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Rubi [A] time = 3.78128, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {3566, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}-\frac{i \sqrt{a-i b} (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i \sqrt{a+i b} (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx &=\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\int \frac{\sqrt{a+b \tan (e+f x)} \left (\frac{1}{2} \left (4 b c^3-3 b c d^2-a d^3\right )+2 b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac{1}{2} d^2 (9 b c-a d) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 b}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\int \frac{-\frac{1}{4} d \left (9 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-14 c d^2\right )\right )+2 b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) \tan (e+f x)+\frac{1}{4} d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 b d}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} d \left (9 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-14 c d^2\right )\right )+2 b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x+\frac{1}{4} d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )}{4 \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 \left (-b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )+b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x\right )}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\operatorname{Subst}\left (\int \frac{-b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )+b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d f}+\frac{\left (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b f}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-i b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-i b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d f}+\frac{\left (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b \tan (e+f x)}\right )}{4 b^2 f}\\ &=\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\left ((i a+b) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{\left ((i a-b) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{\left (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{4 b^2 f}\\ &=\frac{\sqrt{d} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{\left ((i a+b) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\left ((i a-b) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+i b-(c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac{i \sqrt{a-i b} (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i \sqrt{a+i b} (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\sqrt{d} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}\\ \end{align*}
Mathematica [A] time = 5.99989, size = 550, normalized size = 1.62 \[ \frac{\frac{\sqrt{d} \sqrt{c-\frac{a d}{b}} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right )}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}+\frac{4 \left (b d \left (\sqrt{-b^2}-a\right ) \left (d^2-3 c^2\right )+a \sqrt{-b^2} c \left (c^2-3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{b d}{\sqrt{-b^2}}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{b d}{\sqrt{-b^2}}+c}}-\frac{4 \left (-b d \left (a+\sqrt{-b^2}\right ) \left (d^2-3 c^2\right )-a \sqrt{-b^2} c \left (c^2-3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-b^2} d+b c}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{\sqrt{-b^2} d+b c}{b}}}+2 d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}+d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\tan \left ( fx+e \right ) } \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \left (f x + e\right ) + a}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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