Integrand size = 29, antiderivative size = 899 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\frac {2 (a-b) \sqrt {a+b} d \left (6 b c^3-7 a c^2 d-2 b c d^2+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b c^2 \left (3 c^2+3 c d-2 d^2\right )-a d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]
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Time = 2.68 (sec) , antiderivative size = 899, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4027, 3127, 3132, 2890, 3077, 2897, 3075} \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\frac {2 \sqrt {a+b \sec (e+f x)} \sin (e+f x) d^2}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {2 (a-b) \sqrt {a+b} \left (6 b c^3-7 a d c^2-2 b d^2 c+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)} d}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b c^2 \left (3 c^2+3 d c-2 d^2\right )-a d \left (9 c^3-2 d c^2-6 d^2 c+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \]
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Rule 2890
Rule 2897
Rule 3075
Rule 3077
Rule 3127
Rule 3132
Rule 4027
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\cos ^2(e+f x) \sqrt {b+a \cos (e+f x)}}{(d+c \cos (e+f x))^{5/2}} \, dx}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (2 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {-\frac {1}{2} d (3 b c-a d)+\frac {1}{2} \left (3 b c^2-3 a c d-2 b d^2\right ) \cos (e+f x)+\frac {3}{2} a \left (c^2-d^2\right ) \cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{3 c \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (a \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}} \, dx}{c^3 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left (2 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {-\frac {1}{2} c^2 d (3 b c-a d)-\frac {3}{2} a d^2 \left (c^2-d^2\right )+c \left (-3 a d \left (c^2-d^2\right )+\frac {1}{2} c \left (3 b c^2-3 a c d-2 b d^2\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{3 c^3 \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\ & = -\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}-\frac {\left (d \left (6 b c^3-7 a c^2 d-2 b c d^2+3 a d^3\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1+\cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{3 c^2 (c-d) \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left (\left (b c^2 \left (3 c^2+3 c d-2 d^2\right )-a d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}} \, dx}{3 c^3 (c-d) \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 (a-b) \sqrt {a+b} d \left (6 b c^3-7 a c^2 d-2 b c d^2+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b c^2 \left (3 c^2+3 c d-2 d^2\right )-a d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1990\) vs. \(2(899)=1798\).
Time = 7.33 (sec) , antiderivative size = 1990, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\frac {(d+c \cos (e+f x))^3 \sec ^2(e+f x) \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) (d+c \cos (e+f x))^2}-\frac {2 \left (6 b c^3 d \sin (e+f x)-7 a c^2 d^2 \sin (e+f x)-2 b c d^3 \sin (e+f x)+3 a d^4 \sin (e+f x)\right )}{3 c (b c-a d) \left (c^2-d^2\right )^2 (d+c \cos (e+f x))}\right )}{f (c+d \sec (e+f x))^{5/2}}+\frac {(d+c \cos (e+f x))^{5/2} \sec ^2(e+f x) \sqrt {a+b \sec (e+f x)} \left (\frac {4 (b c-a d) \left (3 b^2 c^4-3 a b c^3 d-a^2 c^2 d^2+b^2 c^2 d^2-a b c d^3+a^2 d^4\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}+4 (b c-a d) \left (3 a b c^4-3 a^2 c^3 d+6 b^2 c^3 d-7 a b c^2 d^2-a^2 c d^3-2 b^2 c d^3+4 a b d^4\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {b c-a d}{(a+b) c},\arcsin \left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )+2 \left (6 a b c^3 d-7 a^2 c^2 d^2-2 a b c d^3+3 a^2 d^4\right ) \left (\frac {\sqrt {\frac {-a+b}{a+b}} (a+b) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {d+c \cos (e+f x)} E\left (\arcsin \left (\frac {\sqrt {\frac {-a+b}{a+b}} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {b+a \cos (e+f x)}{a+b}}}\right )|\frac {2 (b c-a d)}{(-a+b) (c+d)}\right )}{a c \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} (e+f x)\right )}{b+a \cos (e+f x)}} \sqrt {b+a \cos (e+f x)} \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {\frac {(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac {2 (b c-a d) \left (\frac {(b c+(a+b) d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {b c-a d}{(a+b) c},\arcsin \left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )}{a c}+\frac {\sqrt {d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt {b+a \cos (e+f x)}}\right )\right )}{3 c (c-d)^2 (c+d)^2 (b c-a d) f \sqrt {b+a \cos (e+f x)} (c+d \sec (e+f x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(18679\) vs. \(2(820)=1640\).
Time = 17.71 (sec) , antiderivative size = 18680, normalized size of antiderivative = 20.78
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a + b \sec {\left (e + f x \right )}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\text {Hanged} \]
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