Optimal. Leaf size=56 \[ \frac{2 \tan (c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left (1,\frac{p+2}{4};\frac{p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
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Rubi [A] time = 0.0415078, antiderivative size = 56, 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} \[ \frac{2 \tan (c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left (1,\frac{p+2}{4};\frac{p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \sqrt{b \tan ^p(c+d x)} \, dx &=\left (\tan ^{-\frac{p}{2}}(c+d x) \sqrt{b \tan ^p(c+d x)}\right ) \int \tan ^{\frac{p}{2}}(c+d x) \, dx\\ &=\frac{\left (\tan ^{-\frac{p}{2}}(c+d x) \sqrt{b \tan ^p(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^{p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{2 \, _2F_1\left (1,\frac{2+p}{4};\frac{6+p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x) \sqrt{b \tan ^p(c+d x)}}{d (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0378778, size = 56, normalized size = 1. \[ \frac{2 \tan (c+d x) \sqrt{b \tan ^p(c+d x)} \, _2F_1\left (1,\frac{p+2}{4};\frac{p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{p}}\, 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 \sqrt{b \tan \left (d x + c\right )^{p}}\,{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 \sqrt{b \tan ^{p}{\left (c + d x \right )}}\, 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 \sqrt{b \tan \left (d x + c\right )^{p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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