Optimal. Leaf size=212 \[ \frac {b^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} d}-\frac {b^{5/2} \log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d}+\frac {b^{5/2} \log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d}+\frac {2 b (b \tan (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.14, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {b^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} d}-\frac {b^{5/2} \log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d}+\frac {b^{5/2} \log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d}+\frac {2 b (b \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rubi steps
\begin {align*} \int (b \tan (c+d x))^{5/2} \, dx &=\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}-b^2 \int \sqrt {b \tan (c+d x)} \, dx\\ &=\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}\\ &=\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {b-x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {b+x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}\\ &=\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}+2 x}{-b-\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}-2 x}{-b+\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{b-\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{b+\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}\\ &=-\frac {b^{5/2} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {b^{5/2} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}-\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}\\ &=\frac {b^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}-\frac {b^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}-\frac {b^{5/2} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {b^{5/2} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {2 b (b \tan (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 40, normalized size = 0.19 \[ -\frac {2 b (b \tan (c+d x))^{3/2} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )-1\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 594, normalized size = 2.80 \[ \frac {12 \, \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \arctan \left (-\frac {b^{10} + \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} b^{7} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \sqrt {\frac {b^{15} \sin \left (d x + c\right ) + \sqrt {\frac {b^{10}}{d^{4}}} b^{10} d^{2} \cos \left (d x + c\right ) + \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {3}{4}} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}}}{b^{10}}\right ) \cos \left (d x + c\right ) + 12 \, \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \arctan \left (\frac {b^{10} - \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} b^{7} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} + \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \sqrt {\frac {b^{15} \sin \left (d x + c\right ) + \sqrt {\frac {b^{10}}{d^{4}}} b^{10} d^{2} \cos \left (d x + c\right ) - \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {3}{4}} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}}}{b^{10}}\right ) \cos \left (d x + c\right ) + 3 \, \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \cos \left (d x + c\right ) \log \left (\frac {b^{15} \sin \left (d x + c\right ) + \sqrt {\frac {b^{10}}{d^{4}}} b^{10} d^{2} \cos \left (d x + c\right ) + \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {3}{4}} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) - 3 \, \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \cos \left (d x + c\right ) \log \left (\frac {b^{15} \sin \left (d x + c\right ) + \sqrt {\frac {b^{10}}{d^{4}}} b^{10} d^{2} \cos \left (d x + c\right ) - \sqrt {2} \left (\frac {b^{10}}{d^{4}}\right )^{\frac {3}{4}} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 8 \, b^{2} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.06, size = 182, normalized size = 0.86 \[ \frac {2 b \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {b^{3} \sqrt {2}\, \ln \left (\frac {b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )}{4 d \left (b^{2}\right )^{\frac {1}{4}}}-\frac {b^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (b^{2}\right )^{\frac {1}{4}}}+\frac {b^{3} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (b^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 176, normalized size = 0.83 \[ -\frac {3 \, b^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right )}{\sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right )}{\sqrt {b}} - \frac {\sqrt {2} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )}{\sqrt {b}} + \frac {\sqrt {2} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )}{\sqrt {b}}\right )} - 8 \, \left (b \tan \left (d x + c\right )\right )^{\frac {3}{2}} b^{2}}{12 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.79, size = 74, normalized size = 0.35 \[ \frac {2\,b\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {{\left (-1\right )}^{1/4}\,b^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,b^{5/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d} \]
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 \[ \int \left (b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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