Optimal. Leaf size=111 \[ -\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{a f}+\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{2} a f}+\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f} \]
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Rubi [A] time = 0.406006, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3566, 3653, 3532, 205, 3634, 63} \[ -\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{a f}+\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{2} a f}+\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3653
Rule 3532
Rule 205
Rule 3634
Rule 63
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^{5/2}}{a+a \tan (e+f x)} \, dx &=\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}+\frac{2 \int \frac{-\frac{a d^3}{2}-\frac{1}{2} a d^3 \tan (e+f x)-\frac{1}{2} a d^3 \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{a}\\ &=\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}+\frac{\int \frac{-\frac{1}{2} a^2 d^3-\frac{1}{2} a^2 d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{a^3}-\frac{1}{2} d^3 \int \frac{1+\tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx\\ &=\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac{\left (a d^6\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a^4 d^6}{2}+d x^2} \, dx,x,\frac{-\frac{1}{2} a^2 d^3+\frac{1}{2} a^2 d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{2 f}\\ &=\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{2} a f}+\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{d}} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{a f}+\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{\sqrt{2} a f}+\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}\\ \end{align*}
Mathematica [A] time = 0.88921, size = 110, normalized size = 0.99 \[ \frac{\left (\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-2 \tan ^{-1}\left (\sqrt{\tan (e+f x)}\right )+4 \sqrt{\tan (e+f x)}\right ) (d \tan (e+f x))^{5/2}}{2 a f \tan ^{\frac{5}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 395, normalized size = 3.6 \begin{align*} 2\,{\frac{{d}^{2}\sqrt{d\tan \left ( fx+e \right ) }}{af}}-{\frac{{d}^{2}\sqrt{2}}{8\,af}\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 ) }-{\frac{{d}^{2}\sqrt{2}}{4\,af}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{d}^{2}\sqrt{2}}{4\,af}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{{d}^{3}\sqrt{2}}{8\,af}\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 ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{{d}^{3}\sqrt{2}}{4\,af}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{{d}^{3}\sqrt{2}}{4\,af}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{1}{af}{d}^{{\frac{5}{2}}}\arctan \left ({\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{d}}}} \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 [A] time = 1.76578, size = 668, normalized size = 6.02 \begin{align*} \left [\frac{\sqrt{2} \sqrt{-d} d^{2} \log \left (\frac{d \tan \left (f x + e\right )^{2} - 2 \, \sqrt{d \tan \left (f x + e\right )}{\left (\sqrt{2} \tan \left (f x + e\right ) - \sqrt{2}\right )} \sqrt{-d} - 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, \sqrt{-d} d^{2} \log \left (\frac{d \tan \left (f x + e\right ) - 2 \, \sqrt{d \tan \left (f x + e\right )} \sqrt{-d} - d}{\tan \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt{d \tan \left (f x + e\right )} d^{2}}{4 \, a f}, -\frac{\sqrt{2} d^{\frac{5}{2}} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}{\left (\sqrt{2} \tan \left (f x + e\right ) - \sqrt{2}\right )}}{2 \, \sqrt{d} \tan \left (f x + e\right )}\right ) + 2 \, d^{\frac{5}{2}} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right ) - 4 \, \sqrt{d \tan \left (f x + e\right )} d^{2}}{2 \, a f}\right ] \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*} \frac{\int \frac{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}{\tan{\left (e + f x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41355, size = 383, normalized size = 3.45 \begin{align*} -\frac{1}{8} \, d^{2}{\left (\frac{2 \, \sqrt{2}{\left (d \sqrt{{\left | d \right |}} +{\left | d \right |}^{\frac{3}{2}}\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 )}{a d f} + \frac{2 \, \sqrt{2}{\left (d \sqrt{{\left | d \right |}} +{\left | d \right |}^{\frac{3}{2}}\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 )}{a d f} + \frac{8 \, \sqrt{d} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right )}{a f} + \frac{\sqrt{2}{\left (d \sqrt{{\left | d \right |}} -{\left | d \right |}^{\frac{3}{2}}\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 )}{a d f} - \frac{\sqrt{2}{\left (d \sqrt{{\left | d \right |}} -{\left | d \right |}^{\frac{3}{2}}\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 )}{a d f} - \frac{16 \, \sqrt{d \tan \left (f x + e\right )}}{a f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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