Optimal. Leaf size=258 \[ \frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}-\frac{i \sqrt{a-i b} (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i \sqrt{a+i b} (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{b} f} \]
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Rubi [A] time = 2.195, antiderivative size = 258, 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 = {3570, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}-\frac{i \sqrt{a-i b} (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i \sqrt{a+i b} (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{b} f} \]
Antiderivative was successfully verified.
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Rule 3570
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))^{3/2} \, dx &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\int \frac{\frac{1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac{1}{2} d (3 b c+a d) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx\\ &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x+\frac{1}{2} d (3 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 )}{f}\\ &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\operatorname{Subst}\left (\int \left (\frac{d (3 b c+a d)}{2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\operatorname{Subst}\left (\int \frac{-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}+\frac{(d (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-2 a c d-b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 a c d+b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}+\frac{(d (3 b c+a d)) \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 f}\\ &=\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\left ((i a+b) (c-i d)^2\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)^2\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{(d (3 b c+a d)) \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 f}\\ &=\frac{\sqrt{d} (3 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 )}{\sqrt{b} f}+\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}+\frac{\left ((i a+b) (c-i d)^2\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)^2\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)^{3/2} \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)^{3/2} \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} (3 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 )}{\sqrt{b} f}+\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}\\ \end{align*}
Mathematica [B] time = 6.09353, size = 1528, normalized size = 5.92 \[ \frac{i (a+i b) \left ((c+i d) \left (\frac{2 (c+i d) \tan ^{-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} \sqrt{-c-i d}}-\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right ) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt{c+d \tan (e+f x)}}\right )-\frac{2 d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )^{3/2} \left (\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right )}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )^{3/2}}+\frac{1}{2 \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )}\right )}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}\right )}{2 f}-\frac{i (i b-a) \left (\frac{2 d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )^{3/2} \left (\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right )}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )^{3/2}}+\frac{1}{2 \left (\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )}+1\right )}\right )}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-(i d-c) \left (\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right ) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 (i d-c) \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{i b-a} \sqrt{c-i d}}\right )\right )}{2 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{3}{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{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \tan{\left (e + f x \right )}} \left (c + d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}\, dx \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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