Optimal. Leaf size=231 \[ -\frac{\left (a^2+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{3/2} d}-\frac{\sqrt{-b+i a} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{b+i a} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 1.59913, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3566, 3647, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ -\frac{\left (a^2+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{3/2} d}-\frac{\sqrt{-b+i a} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{b+i a} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx &=\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\int \frac{\sqrt{a+b \tan (c+d x)} \left (-\frac{a}{2}-2 b \tan (c+d x)-\frac{1}{2} a \tan ^2(c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{2 b}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\int \frac{-\frac{a^2}{4}-2 a b \tan (c+d x)-\frac{1}{4} \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}{2 b}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2}{4}-2 a b x+\frac{1}{4} \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 )}{2 b d}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\operatorname{Subst}\left (\int \left (\frac{-a^2-8 b^2}{4 \sqrt{x} \sqrt{a+b x}}+\frac{2 \left (b^2-a b x\right )}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{b^2-a b x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}-\frac{\left (a^2+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{8 b d}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{\operatorname{Subst}\left (\int \left (\frac{a b+i b^2}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{-a b+i b^2}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}-\frac{\left (a^2+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{4 b d}\\ &=-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}-\frac{(a-i b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{(a+i b) \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 (a^2+8 b^2\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 )}{4 b d}\\ &=-\frac{\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 )}{4 b^{3/2} d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}-\frac{(a-i b) \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{(a+i b) \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{\sqrt{i a-b} \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\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 )}{4 b^{3/2} d}+\frac{\sqrt{i a+b} \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}\\ \end{align*}
Mathematica [A] time = 5.17094, size = 251, normalized size = 1.09 \[ \frac{\frac{\sqrt{b} \sqrt{\tan (c+d x)} \left (a^2+3 a b \tan (c+d x)+2 b^2 \tan ^2(c+d x)\right )-\sqrt{a} \left (a^2+8 b^2\right ) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{3/2} \sqrt{a+b \tan (c+d x)}}+4 \sqrt [4]{-1} \sqrt{-a-i b} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+4 \sqrt [4]{-1} \sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.993, size = 1090940, normalized size = 4722.7 \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 \sqrt{b \tan \left (d x + c\right ) + a} \tan \left (d x + 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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