Optimal. Leaf size=337 \[ \frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{5 a \left (a^2-8 b^2\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{8 \sqrt{b} d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 2.12564, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4241, 3566, 3647, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{5 a \left (a^2-8 b^2\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{8 \sqrt{b} d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3566
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{1}{3} \left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\tan ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} a \left (6 a^2-5 b^2\right )+3 b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac{13}{2} a b^2 \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{\tan (c+d x)} \left (-\frac{39}{4} a^2 b^2+6 a b \left (a^2-3 b^2\right ) \tan (c+d x)+\frac{3}{4} b^2 \left (11 a^2-8 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 b}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{8} a b^2 \left (11 a^2-8 b^2\right )-6 b^3 \left (3 a^2-b^2\right ) \tan (c+d x)+\frac{15}{8} a b^2 \left (a^2-8 b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{6 b^2}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3}{8} a b^2 \left (11 a^2-8 b^2\right )-6 b^3 \left (3 a^2-b^2\right ) x+\frac{15}{8} a b^2 \left (a^2-8 b^2\right ) x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{6 b^2 d}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{15 a b^2 \left (a^2-8 b^2\right )}{8 \sqrt{x} \sqrt{a+b x}}-\frac{6 \left (a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right ) x\right )}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{6 b^2 d}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right ) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 d}+\frac{\left (5 a \left (a^2-8 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{i a b^2 \left (a^2-3 b^2\right )-b^3 \left (3 a^2-b^2\right )}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{i a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right )}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 d}+\frac{\left (5 a \left (a^2-8 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{8 d}\\ &=\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left ((i a-b)^3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left ((i a+b)^3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left (5 a \left (a^2-8 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{8 d}\\ &=\frac{5 a \left (a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{8 \sqrt{b} d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{\left ((i a-b)^3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{\left ((i a+b)^3 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(-a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}\\ &=-\frac{i (i a-b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}+\frac{5 a \left (a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{8 \sqrt{b} d}-\frac{i (i a+b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{\left (11 a^2-8 b^2\right ) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.84435, size = 320, normalized size = 0.95 \[ \frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (3 \left (11 a^2-8 b^2\right ) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{15 a^{3/2} \left (a^2-8 b^2\right ) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+8 b^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}+26 a b \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}-24 (-1)^{3/4} (-a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+24 (-1)^{3/4} (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.245, size = 18215, normalized size = 54.1 \begin{align*} \text{output too large to display} \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 (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cot \left (d x + 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(-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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