Optimal. Leaf size=81 \[ \frac {2 F_1\left (\frac {3}{2};1-m,1;\frac {5}{2};-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-m} \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^m}{3 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 = {3645, 129, 525,
524} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 524
Rule 525
Rule 3645
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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [F]
time = 5.21, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.73, size = 0, normalized size = 0.00 \[\int \left (\sqrt {\tan }\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{m} \sqrt {\tan {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {\mathrm {tan}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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