Optimal. Leaf size=269 \[ -\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}-\frac{a^2 d^{5/2} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{a^2 d^{5/2} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f} \]
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Rubi [A] time = 0.25824, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3543, 12, 16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}-\frac{a^2 d^{5/2} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{a^2 d^{5/2} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 12
Rule 16
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx &=\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\int 2 a^2 \tan (e+f x) (d \tan (e+f x))^{5/2} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\left (2 a^2\right ) \int \tan (e+f x) (d \tan (e+f x))^{5/2} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{\left (2 a^2\right ) \int (d \tan (e+f x))^{7/2} \, dx}{d}\\ &=\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}-\left (2 a^2 d\right ) \int (d \tan (e+f x))^{3/2} \, dx\\ &=-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\left (2 a^2 d^3\right ) \int \frac{1}{\sqrt{d \tan (e+f x)}} \, dx\\ &=-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{\left (2 a^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{\left (4 a^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{\left (2 a^2 d^3\right ) \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}+\frac{\left (2 a^2 d^3\right ) \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}-\frac{\left (a^2 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{\left (a^2 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{\left (a^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}+\frac{\left (a^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{a^2 d^{5/2} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{a^2 d^{5/2} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{\left (\sqrt{2} a^2 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\left (\sqrt{2} a^2 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}\\ &=-\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{a^2 d^{5/2} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{a^2 d^{5/2} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\\ \end{align*}
Mathematica [A] time = 1.26283, size = 187, normalized size = 0.7 \[ \frac{a^2 (d \tan (e+f x))^{5/2} \left (20 \tan ^{\frac{7}{2}}(e+f x)+56 \tan ^{\frac{5}{2}}(e+f x)-70 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right )+70 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-280 \sqrt{\tan (e+f x)}-35 \sqrt{2} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )+35 \sqrt{2} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )}{70 f \tan ^{\frac{5}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 234, normalized size = 0.9 \begin{align*}{\frac{2\,{a}^{2}}{7\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{a}^{2}}{5\,f} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}-4\,{\frac{{a}^{2}{d}^{2}\sqrt{d\tan \left ( fx+e \right ) }}{f}}+{\frac{{a}^{2}{d}^{2}\sqrt{2}}{f}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{{a}^{2}{d}^{2}\sqrt{2}}{f}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{a}^{2}{d}^{2}\sqrt{2}}{2\,f}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) } \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 = 1.91268, size = 1797, normalized size = 6.68 \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*} a^{2} \left (\int \left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}\, dx + \int 2 \left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}} \tan{\left (e + f x \right )}\, dx + \int \left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}} \tan ^{2}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28562, size = 396, normalized size = 1.47 \begin{align*} \frac{\sqrt{2} a^{2} d^{2} \sqrt{{\left | d \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{f} + \frac{\sqrt{2} a^{2} d^{2} \sqrt{{\left | d \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{f} + \frac{\sqrt{2} a^{2} d^{2} \sqrt{{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) + \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{2 \, f} - \frac{\sqrt{2} a^{2} d^{2} \sqrt{{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) - \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{2 \, f} + \frac{2 \,{\left (5 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{9} f^{6} \tan \left (f x + e\right )^{3} + 14 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{9} f^{6} \tan \left (f x + e\right )^{2} - 70 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{9} f^{6}\right )}}{35 \, d^{7} f^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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