Optimal. Leaf size=145 \[ \frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
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Rubi [A] time = 0.17, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3670, 446, 88, 50, 63, 208} \[ \frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5 \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{3/2}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {(-a-b) (a+b x)^{3/2}}{b}+\frac {(a+b x)^{3/2}}{1+x}+\frac {(a+b x)^{5/2}}{b}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}+\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}+\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{b f}\\ &=-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2 f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2 f}\\ \end {align*}
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Mathematica [A] time = 1.39, size = 139, normalized size = 0.96 \[ \frac {\frac {2 \left (a+b \tan ^2(e+f x)\right )^{7/2}}{7 b^2}-\frac {2 (a+b) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b^2}+\frac {2}{3} \left (a+b \tan ^2(e+f x)\right )^{3/2}+2 (a-b) \left (\sqrt {a+b \tan ^2(e+f x)}-\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 414, normalized size = 2.86 \[ \left [-\frac {105 \, {\left (a b^{2} - b^{3}\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (15 \, b^{3} \tan \left (f x + e\right )^{6} + 3 \, {\left (8 \, a b^{2} - 7 \, b^{3}\right )} \tan \left (f x + e\right )^{4} - 6 \, a^{3} - 21 \, a^{2} b + 140 \, a b^{2} - 105 \, b^{3} + {\left (3 \, a^{2} b - 42 \, a b^{2} + 35 \, b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{420 \, b^{2} f}, \frac {105 \, {\left (a b^{2} - b^{3}\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \, {\left (15 \, b^{3} \tan \left (f x + e\right )^{6} + 3 \, {\left (8 \, a b^{2} - 7 \, b^{3}\right )} \tan \left (f x + e\right )^{4} - 6 \, a^{3} - 21 \, a^{2} b + 140 \, a b^{2} - 105 \, b^{3} + {\left (3 \, a^{2} b - 42 \, a b^{2} + 35 \, b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{210 \, b^{2} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 196, normalized size = 1.35 \[ \frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b} f} + \frac {15 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {7}{2}} b^{12} f^{6} - 21 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} a b^{12} f^{6} - 21 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} b^{13} f^{6} + 35 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b^{14} f^{6} + 105 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} a b^{14} f^{6} - 105 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b^{15} f^{6}}{105 \, b^{14} f^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 256, normalized size = 1.77 \[ \frac {\left (\tan ^{2}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{7 f b}-\frac {2 a \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{35 f \,b^{2}}-\frac {\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{5 b f}+\frac {b \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}+\frac {4 a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}-\frac {b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{f}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}} \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 40.84, size = 233, normalized size = 1.61 \[ \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{7/2}}{7\,b^2\,f}-\left (\frac {2\,a}{5\,b^2\,f}-\frac {a-b}{5\,b^2\,f}\right )\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}-\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )\,\left (\left (\frac {2\,a}{b^2\,f}-\frac {a-b}{b^2\,f}\right )\,\left (a-b\right )-\frac {a^2}{b^2\,f}\right )-{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}\,\left (\frac {\left (\frac {2\,a}{b^2\,f}-\frac {a-b}{b^2\,f}\right )\,\left (a-b\right )}{3}-\frac {a^2}{3\,b^2\,f}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{f} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{5}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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