Optimal. Leaf size=429 \[ \frac{\left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt{d} f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}-\frac{i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f} \]
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Rubi [A] time = 5.46731, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{\left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt{d} f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}-\frac{i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f} \]
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 (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx &=\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\int \frac{(a+b \tan (e+f x))^{3/2} \left (\frac{1}{2} \left (6 b c^3-5 b c d^2-a d^3\right )+3 b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac{1}{2} d^2 (13 b c-a d) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{3 b}\\ &=\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\int \frac{\sqrt{a+b \tan (e+f x)} \left (-\frac{3}{4} d \left (13 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-10 c d^2\right )\right )+6 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{3}{4} d^2 \left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{6 b d}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\int \frac{-\frac{3}{8} d^2 \left (a^3 d^3+b^3 c \left (11 c^2-8 d^2\right )+a b^2 d \left (51 c^2-8 d^2\right )-a^2 b \left (16 c^3-33 c d^2\right )\right )+6 b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)+\frac{3}{8} d^2 \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{6 b d^2}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{8} d^2 \left (a^3 d^3+b^3 c \left (11 c^2-8 d^2\right )+a b^2 d \left (51 c^2-8 d^2\right )-a^2 b \left (16 c^3-33 c d^2\right )\right )+6 b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x+\frac{3}{8} d^2 \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c 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 )}{6 b d^2 f}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 d^2 \left (15 a^2 b c d^2-a^3 d^3+5 b^3 \left (c^3-8 c d^2\right )+a b^2 \left (45 c^2 d-24 d^3\right )\right )}{8 \sqrt{a+b x} \sqrt{c+d x}}+\frac{6 \left (-b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x\right )}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{6 b d^2 f}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\operatorname{Subst}\left (\int \frac{-b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d^2 f}+\frac{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{16 b f}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-i b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )-b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{-i b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d^2 f}+\frac{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\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 )}{8 b^2 f}\\ &=\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}-\frac{\left ((a+i b)^2 (i c-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 ((a-i b)^2 (i c+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 (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\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 )}{8 b^2 f}\\ &=\frac{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c 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 )}{8 b^{3/2} \sqrt{d} f}+\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}-\frac{\left ((a+i b)^2 (i c-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 ((a-i b)^2 (i c+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 (a-i b)^{3/2} (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 (a+i b)^{3/2} (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{\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c 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 )}{8 b^{3/2} \sqrt{d} f}+\frac{\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}\\ \end{align*}
Mathematica [A] time = 8.10674, size = 773, normalized size = 1.8 \[ \frac{\frac{\frac{3 d \left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 f}+\frac{-\frac{6 b d^2 \left (\sqrt{-b^2} \left (a^2 \left (-\left (c^3-3 c d^2\right )\right )+a b \left (6 c^2 d-2 d^3\right )+b^2 c \left (c^2-3 d^2\right )\right )-b \left (a^2 \left (3 c^2 d-d^3\right )+2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{b d}{\sqrt{-b^2}}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{b d}{\sqrt{-b^2}}+c}}-\frac{6 b d^2 \left (\sqrt{-b^2} \left (a^2 \left (-\left (c^3-3 c d^2\right )\right )+a b \left (6 c^2 d-2 d^3\right )+b^2 c \left (c^2-3 d^2\right )\right )+b \left (a^2 \left (3 c^2 d-d^3\right )+2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-b^2} d+b c}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{\sqrt{-b^2} d+b c}{b}}}+\frac{3 \sqrt{b} d^{3/2} \sqrt{c-\frac{a d}{b}} \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right )}{4 \sqrt{c+d \tan (e+f x)}}}{b d f}}{2 d}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{4 f}}{3 b}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \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{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\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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