Integrand size = 29, antiderivative size = 731 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {3 b \sqrt {3+b} (b c-9 d) (c-d) \sqrt {c+d} E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 (b c-3 d) d^2 f}-\frac {\sqrt {c+d} \left (30 b c d-135 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 \sqrt {3+b} d^3 f}+\frac {3 b^2 (b c-9 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {3+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}-\frac {(3+b)^{3/2} (3 b c-21 d-2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 d^2 \sqrt {c+d} f} \]
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Time = 1.42 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2872, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{4 d^3 f \sqrt {a+b}}-\frac {(a+b)^{3/2} (-7 a d+3 b c-2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{4 d^2 f \sqrt {c+d}}-\frac {3 b \sqrt {a+b} (c-d) \sqrt {c+d} (b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{4 d^2 f (b c-a d)}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f} \]
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Rule 2872
Rule 2890
Rule 2897
Rule 3075
Rule 3077
Rule 3132
Rule 3140
Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {\int \frac {\frac {1}{2} \left (b^3 c+4 a^3 d+a b^2 d\right )-b \left (a b c-6 a^2 d-b^2 d\right ) \sin (e+f x)-\frac {3}{2} b^2 (b c-3 a d) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{2 d} \\ & = \frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {\int \frac {\frac {1}{2} \left (3 b^4 c^2-10 a b^3 c d+8 a^4 d^2+11 a^2 b^2 d^2\right )-b \left (11 a^2 b c d-b^3 c d-16 a^3 d^2-3 a b^2 \left (c^2+d^2\right )\right ) \sin (e+f x)-\frac {1}{2} b^2 \left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{4 d^2} \\ & = \frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {\int \frac {\frac {1}{2} b^2 \left (3 b^4 c^2-10 a b^3 c d+8 a^4 d^2+11 a^2 b^2 d^2\right )+\frac {1}{2} a^2 b^2 \left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right )+b \left (-b^2 \left (11 a^2 b c d-b^3 c d-16 a^3 d^2-3 a b^2 \left (c^2+d^2\right )\right )+a b^2 \left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{4 b^2 d^2}-\frac {\left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 d^2} \\ & = -\frac {\sqrt {c+d} \left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 \sqrt {a+b} d^3 f}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {(3 b (a+b) (b c-3 a d) (b c-a d)) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{8 d^2}-\frac {((a+b) (b c-a d) (3 b c-7 a d-2 b d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 d^2} \\ & = -\frac {3 b \sqrt {a+b} (c-d) \sqrt {c+d} (b c-3 a d) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 d^2 (b c-a d) f}-\frac {\sqrt {c+d} \left (10 a b c d-15 a^2 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 \sqrt {a+b} d^3 f}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}-\frac {(a+b)^{3/2} (3 b c-7 a d-2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 d^2 \sqrt {c+d} f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1865\) vs. \(2(731)=1462\).
Time = 21.11 (sec) , antiderivative size = 1865, normalized size of antiderivative = 2.55 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}+\frac {-\frac {4 (-b c+3 d) \left (-b^3 c+216 d+33 b^2 d\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+3 d) \left (-12 b^2 c+216 b d+4 b^3 d\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 \left (3 b^3 c-27 b^2 d\right ) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {3+b \sin (e+f x)}}+\frac {\sqrt {\frac {3-b}{3+b}} (3+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {3-b}{3+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {3+b \sin (e+f x)}{3+b}}}\right )|\frac {2 (-b c+3 d)}{(3-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(3+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{3+b \sin (e+f x)}} \sqrt {3+b \sin (e+f x)} \sqrt {\frac {3+b \sin (e+f x)}{3+b}} \sqrt {\frac {(3+b) (c+d \sin (e+f x))}{(c+d) (3+b \sin (e+f x))}}}-\frac {2 (-b c+3 d) \left (\frac {((3+b) c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{8 d f} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 21.70 (sec) , antiderivative size = 656608, normalized size of antiderivative = 898.23
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\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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