Optimal. Leaf size=181 \[ \frac{i \sqrt{-b+i a} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{i \sqrt{b+i a} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.348561, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3568, 3649, 21, 3575, 910, 93, 205, 208} \[ \frac{i \sqrt{-b+i a} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{i \sqrt{b+i a} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3568
Rule 3649
Rule 21
Rule 3575
Rule 910
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \tan (c+d x)}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2}{3} \int \frac{-\frac{b}{2}+\frac{3}{2} a \tan (c+d x)+b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{4 \int \frac{-\frac{3 a^2}{4}-\frac{3}{4} a b \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\int \frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \left (\frac{i a-b}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{i a+b}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\frac{(i a-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{(i a+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{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\frac{(i a-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{(i a+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{i \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{i \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{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.535467, size = 161, normalized size = 0.89 \[ \frac{-\frac{2 (a+b \tan (c+d x))^{3/2}}{a \tan ^{\frac{3}{2}}(c+d x)}-3 (-1)^{3/4} \sqrt{-a-i b} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+3 (-1)^{3/4} \sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.319, size = 1089917, normalized size = 6021.6 \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{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \tan{\left (c + d x \right )}}}{\tan ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx \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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