Optimal. Leaf size=71 \[ \frac {2 b^2 \, _2F_1\left (1,\frac {1}{4} (2+5 n);\frac {1}{4} (6+5 n);-\tan ^2(e+f x)\right ) \tan ^{1+2 n}(e+f x) \sqrt {b \tan ^n(e+f x)}}{f (2+5 n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 = {3740, 3557,
371} \begin {gather*} \frac {2 b^2 \tan ^{2 n+1}(e+f x) \sqrt {b \tan ^n(e+f x)} \, _2F_1\left (1,\frac {1}{4} (5 n+2);\frac {1}{4} (5 n+6);-\tan ^2(e+f x)\right )}{f (5 n+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \left (b \tan ^n(e+f x)\right )^{5/2} \, dx &=\left (b^2 \tan ^{-\frac {n}{2}}(e+f x) \sqrt {b \tan ^n(e+f x)}\right ) \int \tan ^{\frac {5 n}{2}}(e+f x) \, dx\\ &=\frac {\left (b^2 \tan ^{-\frac {n}{2}}(e+f x) \sqrt {b \tan ^n(e+f x)}\right ) \text {Subst}\left (\int \frac {x^{5 n/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 b^2 \, _2F_1\left (1,\frac {1}{4} (2+5 n);\frac {1}{4} (6+5 n);-\tan ^2(e+f x)\right ) \tan ^{1+2 n}(e+f x) \sqrt {b \tan ^n(e+f x)}}{f (2+5 n)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 62, normalized size = 0.87 \begin {gather*} \frac {2 \, _2F_1\left (1,\frac {1}{4} (2+5 n);\frac {1}{4} (6+5 n);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^n(e+f x)\right )^{5/2}}{f (2+5 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (b \left (\tan ^{n}\left (f x +e \right )\right )\right )^{\frac {5}{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (b \tan ^{n}{\left (e + f x \right )}\right )^{\frac {5}{2}}\, 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 {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^n\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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