Optimal. Leaf size=81 \[ \frac{2 \tan ^{\frac{3}{2}}(c+d x) (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left (\frac{3}{2};1-m,1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{3 d} \]
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Rubi [A] time = 0.117413, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3564, 130, 511, 510} \[ \frac{2 \tan ^{\frac{3}{2}}(c+d x) (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left (\frac{3}{2};1-m,1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 3564
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^m \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{-\frac{i x}{a}} (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+i a x^2\right )^{-1+m}}{-a^2+i a^2 x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{\left (2 a^2 (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (1+i x^2\right )^{-1+m}}{-a^2+i a^2 x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{2 F_1\left (\frac{3}{2};1-m,1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-m} \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^m}{3 d}\\ \end{align*}
Mathematica [F] time = 4.88459, size = 0, normalized size = 0. \[ \int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.237, size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan \left ( dx+c \right ) } \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{m}\, 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 (i \, a \tan \left (d x + c\right ) + a\right )}^{m} \sqrt{\tan \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} \sqrt{\frac{-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}, x\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*} \int \left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{m} \sqrt{\tan{\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{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m} \sqrt{\tan \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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