Optimal. Leaf size=246 \[ \frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}-\frac{a^2 d^{3/2} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{a^2 d^{3/2} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.227639, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 15, 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, 297, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}-\frac{a^2 d^{3/2} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{a^2 d^{3/2} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 12
Rule 16
Rule 3473
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2 \, dx &=\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\int 2 a^2 \tan (e+f x) (d \tan (e+f x))^{3/2} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\left (2 a^2\right ) \int \tan (e+f x) (d \tan (e+f x))^{3/2} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{\left (2 a^2\right ) \int (d \tan (e+f x))^{5/2} \, dx}{d}\\ &=\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}-\left (2 a^2 d\right ) \int \sqrt{d \tan (e+f x)} \, dx\\ &=\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}-\frac{\left (2 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}-\frac{\left (4 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{\left (2 a^2 d^2\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^2\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 \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}-\frac{\left (a^2 d^{3/2}\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^{3/2}\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^2\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^2\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^{3/2} \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^{3/2} \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 \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}-\frac{\left (\sqrt{2} a^2 d^{3/2}\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^{3/2}\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^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{a^2 d^{3/2} \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^{3/2} \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 \tan (e+f x))^{3/2}}{3 f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}\\ \end{align*}
Mathematica [C] time = 0.374567, size = 52, normalized size = 0.21 \[ \frac{2 a^2 (d \tan (e+f x))^{3/2} \left (-10 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )+3 \tan (e+f x)+10\right )}{15 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 213, normalized size = 0.9 \begin{align*}{\frac{2\,{a}^{2}}{5\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{4\,{a}^{2}}{3\,f} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}\sqrt{2}}{2\,f}\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{{a}^{2}{d}^{2}\sqrt{2}}{f}\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{{a}^{2}{d}^{2}\sqrt{2}}{f}\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}}}}} \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 = 1.89997, size = 1769, normalized size = 7.19 \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{3}{2}}\, dx + \int 2 \left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}} \tan{\left (e + f x \right )}\, dx + \int \left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{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 = 1.29365, size = 370, normalized size = 1.5 \begin{align*} -\frac{1}{30} \,{\left (\frac{30 \, \sqrt{2} a^{2}{\left | d \right |}^{\frac{3}{2}} \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 )}{d f} + \frac{30 \, \sqrt{2} a^{2}{\left | d \right |}^{\frac{3}{2}} \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 )}{d f} - \frac{15 \, \sqrt{2} a^{2}{\left | d \right |}^{\frac{3}{2}} \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 )}{d f} + \frac{15 \, \sqrt{2} a^{2}{\left | d \right |}^{\frac{3}{2}} \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 )}{d f} - \frac{4 \,{\left (3 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{10} f^{4} \tan \left (f x + e\right )^{2} + 10 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{10} f^{4} \tan \left (f x + e\right )\right )}}{d^{10} f^{5}}\right )} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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