Optimal. Leaf size=286 \[ \frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}-\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.126066, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}-\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \left (b \tan ^3(c+d x)\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{9}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}-\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{5}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \int \sqrt{\tan (c+d x)} \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}+\frac{\left (2 b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}-\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}+\frac{b \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\left (b \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \tan ^3(c+d x)}}{3 d}-\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \tan ^2(c+d x) \sqrt{b \tan ^3(c+d x)}}{7 d}\\ \end{align*}
Mathematica [C] time = 0.0606522, size = 54, normalized size = 0.19 \[ \frac{2 b \sqrt{b \tan ^3(c+d x)} \left (7 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right )+3 \tan ^2(c+d x)-7\right )}{21 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 235, normalized size = 0.8 \begin{align*}{\frac{1}{84\,d \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{2}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 24\, \left ( b\tan \left ( dx+c \right ) \right ) ^{7/2}\sqrt [4]{{b}^{2}}+21\,{b}^{4}\sqrt{2}\ln \left ( -{\frac{\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}-b\tan \left ( dx+c \right ) -\sqrt{{b}^{2}}}{b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}}}} \right ) +42\,{b}^{4}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) -42\,{b}^{4}\sqrt{2}\arctan \left ({\frac{-\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) -56\, \left ( b\tan \left ( dx+c \right ) \right ) ^{3/2}{b}^{2}\sqrt [4]{{b}^{2}} \right ) \left ( b\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43031, size = 189, normalized size = 0.66 \begin{align*} \frac{24 \, b^{\frac{3}{2}} \tan \left (d x + c\right )^{\frac{7}{2}} - 56 \, b^{\frac{3}{2}} \tan \left (d x + c\right )^{\frac{3}{2}} + 21 \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} b^{\frac{3}{2}}}{84 \, d} \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 \left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34921, size = 342, normalized size = 1.2 \begin{align*} \frac{1}{84} \, b{\left (\frac{42 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} + 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b d} + \frac{42 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} - 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b d} - \frac{21 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b d} + \frac{21 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b d} + \frac{8 \,{\left (3 \, \sqrt{b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )^{3} - 7 \, \sqrt{b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )\right )}}{b^{21} d^{7}}\right )} \mathrm{sgn}\left (\tan \left (d x + c\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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