Optimal. Leaf size=65 \[ \frac{2 b \tan ^{n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (3 n+2),\frac{3 (n+2)}{4},-\tan ^2(e+f x)\right )}{f (3 n+2)} \]
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Rubi [A] time = 0.0447993, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac{2 b \tan ^{n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left (1,\frac{1}{4} (3 n+2);\frac{3 (n+2)}{4};-\tan ^2(e+f x)\right )}{f (3 n+2)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (b \tan ^n(e+f x)\right )^{3/2} \, dx &=\left (b \tan ^{-\frac{n}{2}}(e+f x) \sqrt{b \tan ^n(e+f x)}\right ) \int \tan ^{\frac{3 n}{2}}(e+f x) \, dx\\ &=\frac{\left (b \tan ^{-\frac{n}{2}}(e+f x) \sqrt{b \tan ^n(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^{3 n/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 b \, _2F_1\left (1,\frac{1}{4} (2+3 n);\frac{3 (2+n)}{4};-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x) \sqrt{b \tan ^n(e+f x)}}{f (2+3 n)}\\ \end{align*}
Mathematica [A] time = 0.0683063, size = 60, normalized size = 0.92 \[ \frac{2 \tan (e+f x) \left (b \tan ^n(e+f x)\right )^{3/2} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (3 n+2),\frac{3 (n+2)}{4},-\tan ^2(e+f x)\right )}{f (3 n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )^{n}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 ^{n}{\left (e + f x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tan \left (f x + e\right )^{n}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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