Optimal. Leaf size=264 \[ \frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{a-i b}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{a+i b}} \]
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Rubi [A] time = 2.50004, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {3566, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{a-i b}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{a+i b}} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2}}{\sqrt{a+b \tan (e+f x)}} \, dx &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}+\frac{\int \frac{\frac{1}{2} \left (2 b c^3-d^2 (b c+a d)\right )+b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac{1}{2} d^2 (5 b c-a d) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{b}\\ &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (2 b c^3-d^2 (b c+a d)\right )+b d \left (3 c^2-d^2\right ) x+\frac{1}{2} d^2 (5 b c-a d) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b f}\\ &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{d^2 (5 b c-a d)}{2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{b f}\\ &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}+\frac{\operatorname{Subst}\left (\int \frac{b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b f}+\frac{\left (d^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b f}\\ &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{i b c \left (c^2-3 d^2\right )-b d \left (3 c^2-d^2\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{i b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b f}+\frac{\left (d^2 (5 b c-a d)\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 )}{b^2 f}\\ &=\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}-\frac{(i c-d)^3 \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{(i c+d)^3 \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^2 (5 b c-a d)\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 )}{b^2 f}\\ &=\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}-\frac{(i c-d)^3 \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 c+d)^3 \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 (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 )}{\sqrt{a-i b} f}+\frac{i (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 )}{\sqrt{a+i b} f}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}\\ \end{align*}
Mathematica [A] time = 2.42881, size = 432, normalized size = 1.64 \[ \frac{\frac{b \left (\sqrt{-b^2} c \left (c^2-3 d^2\right )-b d \left (d^2-3 c^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{b d}{\sqrt{-b^2}}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{b d}{\sqrt{-b^2}}+c}}+\frac{b \left (\sqrt{-b^2} c \left (c^2-3 d^2\right )+b d \left (d^2-3 c^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-b^2} d+b c}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{\sqrt{-b^2} d+b c}{b}}}+b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}+\frac{\sqrt{b} d^{3/2} (5 b c-a d) \sqrt{c-\frac{a d}{b}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right )}{\sqrt{c+d \tan (e+f x)}}}{b^2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{a+b\tan \left ( fx+e \right ) }}}}\, 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 \frac{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{\sqrt{b \tan \left (f x + e\right ) + a}}\,{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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