Optimal. Leaf size=364 \[ -\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}-\frac {b^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b^2 \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f} \]
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Rubi [A] time = 0.15, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {b^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b^2 \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \left (b \tan ^3(e+f x)\right )^{5/2} \, dx &=\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {15}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)}\\ &=\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}-\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {11}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {7}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)}\\ &=\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}-\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \int \tan ^{\frac {3}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \int \frac {1}{\sqrt {\tan (e+f x)}} \, dx}{\tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f \tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (2 b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {b^2 \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}+\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\left (b^2 \sqrt {b \tan ^3(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}\\ &=-\frac {2 b^2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {b^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {b^2 \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {b^2 \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 b^2 \tan (e+f x) \sqrt {b \tan ^3(e+f x)}}{5 f}-\frac {2 b^2 \tan ^3(e+f x) \sqrt {b \tan ^3(e+f x)}}{9 f}+\frac {2 b^2 \tan ^5(e+f x) \sqrt {b \tan ^3(e+f x)}}{13 f}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 199, normalized size = 0.55 \[ \frac {b \left (b \tan ^3(e+f x)\right )^{3/2} \left (-1170 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )+1170 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )+360 \tan ^{\frac {13}{2}}(e+f x)-520 \tan ^{\frac {9}{2}}(e+f x)+936 \tan ^{\frac {5}{2}}(e+f x)-4680 \sqrt {\tan (e+f x)}-585 \sqrt {2} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )+585 \sqrt {2} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )\right )}{2340 f \tan ^{\frac {9}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 263, normalized size = 0.72 \[ \frac {\left (b \left (\tan ^{3}\left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (360 \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}}-520 b^{2} \left (b \tan \left (f x +e \right )\right )^{\frac {9}{2}}+585 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (f x +e \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+936 \left (b \tan \left (f x +e \right )\right )^{\frac {5}{2}} b^{4}-4680 b^{6} \sqrt {b \tan \left (f x +e \right )}\right )}{2340 f \tan \left (f x +e \right )^{5} \left (b \tan \left (f x +e \right )\right )^{\frac {5}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 178, normalized size = 0.49 \[ \frac {360 \, b^{\frac {5}{2}} \tan \left (f x + e\right )^{\frac {13}{2}} - 520 \, b^{\frac {5}{2}} \tan \left (f x + e\right )^{\frac {9}{2}} + 936 \, b^{\frac {5}{2}} \tan \left (f x + e\right )^{\frac {5}{2}} + 585 \, {\left (2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - \sqrt {2} \sqrt {b} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )} b^{2} - 4680 \, b^{\frac {5}{2}} \sqrt {\tan \left (f x + e\right )}}{2340 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}^{5/2} \,d x \]
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 \[ \int \left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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