Optimal. Leaf size=212 \[ \frac{b^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} d}-\frac{b^{5/2} \log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{b^{5/2} \log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{2 b (b \tan (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.144697, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{b^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} d}-\frac{b^{5/2} \log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{b^{5/2} \log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{2 b (b \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int (b \tan (c+d x))^{5/2} \, dx &=\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}-b^2 \int \sqrt{b \tan (c+d x)} \, dx\\ &=\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac{b^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}+2 x}{-b-\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{b^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}-2 x}{-b+\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}\\ &=-\frac{b^{5/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{b^{5/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac{b^{5/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}+\frac{b^{5/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}\\ &=\frac{b^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{5/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{b^{5/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{2 b (b \tan (c+d x))^{3/2}}{3 d}\\ \end{align*}
Mathematica [C] time = 0.0683589, size = 40, normalized size = 0.19 \[ -\frac{2 b (b \tan (c+d x))^{3/2} \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right )-1\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 182, normalized size = 0.9 \begin{align*}{\frac{2\,b}{3\,d} \left ( b\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{3}\sqrt{2}}{4\,d}\ln \left ({ \left ( b\tan \left ( dx+c \right ) -\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) \left ( b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}-{\frac{{b}^{3}\sqrt{2}}{2\,d}\arctan \left ({\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}+{\frac{{b}^{3}\sqrt{2}}{2\,d}\arctan \left ( -{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71795, size = 1521, normalized size = 7.17 \begin{align*} \text{result too large to display} \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{\left (c + d x \right )}\right )^{\frac{5}{2}}\, dx \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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