Optimal. Leaf size=114 \[ \frac {2^{n+\frac {5}{2}} \tan ^{\frac {3}{2}}(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n+\frac {3}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {3}{4};n+\frac {1}{2},1;\frac {7}{4};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3889} \[ \frac {2^{n+\frac {5}{2}} \tan ^{\frac {3}{2}}(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n+\frac {3}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {3}{4};n+\frac {1}{2},1;\frac {7}{4};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 3889
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx &=\frac {2^{\frac {5}{2}+n} F_1\left (\frac {3}{4};\frac {1}{2}+n,1;\frac {7}{4};-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{\frac {3}{2}+n} (a+a \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 2.23, size = 238, normalized size = 2.09 \[ \frac {56 \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right ) \sqrt {\tan (c+d x)} (a (\sec (c+d x)+1))^n F_1\left (\frac {3}{4};n+\frac {1}{2},1;\frac {7}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d \left (6 (\cos (c+d x)-1) \left (2 F_1\left (\frac {7}{4};n+\frac {1}{2},2;\frac {11}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-(2 n+1) F_1\left (\frac {7}{4};n+\frac {3}{2},1;\frac {11}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )+21 (\cos (c+d x)+1) F_1\left (\frac {3}{4};n+\frac {1}{2},1;\frac {7}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\sqrt {\tan }\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\mathrm {tan}\left (c+d\,x\right )}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \sqrt {\tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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