Optimal. Leaf size=121 \[ \frac {a \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(e+f x)\right )}{f (m+1)}+\frac {2 b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);-\tan ^2(e+f x)\right )}{c f (2 m+3)} \]
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Rubi [A] time = 0.33, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3670, 15, 1831, 364} \[ \frac {a \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(e+f x)\right )}{f (m+1)}+\frac {2 b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);-\tan ^2(e+f x)\right )}{c f (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 364
Rule 1831
Rule 3670
Rubi steps
\begin {align*} \int (d \tan (e+f x))^m \left (a+b \sqrt {c \tan (e+f x)}\right ) \, dx &=\frac {c \operatorname {Subst}\left (\int \frac {\left (a+b \sqrt {x}\right ) \left (\frac {d x}{c}\right )^m}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac {(2 c) \operatorname {Subst}\left (\int \frac {x \left (\frac {d x^2}{c}\right )^m (a+b x)}{c^2+x^4} \, dx,x,\sqrt {c \tan (e+f x)}\right )}{f}\\ &=\frac {\left (2 c (c \tan (e+f x))^{-m} (d \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {x^{1+2 m} (a+b x)}{c^2+x^4} \, dx,x,\sqrt {c \tan (e+f x)}\right )}{f}\\ &=\frac {\left (2 c (c \tan (e+f x))^{-m} (d \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \left (\frac {a x^{1+2 m}}{c^2+x^4}+\frac {b x^{2+2 m}}{c^2+x^4}\right ) \, dx,x,\sqrt {c \tan (e+f x)}\right )}{f}\\ &=\frac {\left (2 a c (c \tan (e+f x))^{-m} (d \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {x^{1+2 m}}{c^2+x^4} \, dx,x,\sqrt {c \tan (e+f x)}\right )}{f}+\frac {\left (2 b c (c \tan (e+f x))^{-m} (d \tan (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {x^{2+2 m}}{c^2+x^4} \, dx,x,\sqrt {c \tan (e+f x)}\right )}{f}\\ &=\frac {a \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m}{f (1+m)}+\frac {2 b \, _2F_1\left (1,\frac {1}{4} (3+2 m);\frac {1}{4} (7+2 m);-\tan ^2(e+f x)\right ) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m}{c f (3+2 m)}\\ \end {align*}
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Mathematica [C] time = 0.65, size = 304, normalized size = 2.51 \[ \frac {\tan (e+f x) (d \tan (e+f x))^m \left (\left (a-b \sqrt [4]{-c^2}\right ) \, _2F_1\left (1,2 (m+1);2 m+3;-\frac {\sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )+\left (a+i b \sqrt [4]{-c^2}\right ) \, _2F_1\left (1,2 (m+1);2 m+3;-\frac {i \sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )+a \, _2F_1\left (1,2 (m+1);2 m+3;\frac {i \sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )+a \, _2F_1\left (1,2 (m+1);2 m+3;\frac {\sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )-i b \sqrt [4]{-c^2} \, _2F_1\left (1,2 (m+1);2 m+3;\frac {i \sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )+b \sqrt [4]{-c^2} \, _2F_1\left (1,2 (m+1);2 m+3;\frac {\sqrt {c \tan (e+f x)}}{\sqrt [4]{-c^2}}\right )\right )}{4 f (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \tan \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{m} b + \left (d \tan \left (f x + e\right )\right )^{m} a, x\right ) \]
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 \[ \int {\left (\sqrt {c \tan \left (f x + e\right )} b + a\right )} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.33, size = 0, normalized size = 0.00 \[ \int \left (a +b \sqrt {c \tan \left (f x +e \right )}\right ) \left (d \tan \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (\sqrt {c \tan \left (f x + e\right )} b + a\right )} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int \left (a+b\,\sqrt {c\,\mathrm {tan}\left (e+f\,x\right )}\right )\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^m \,d x \]
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 \[ \int \left (d \tan {\left (e + f x \right )}\right )^{m} \left (a + b \sqrt {c \tan {\left (e + f x \right )}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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