Optimal. Leaf size=784 \[ \frac {\sqrt {a+b} (c-d) \sqrt {c+d} (5 b c+a d) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 b (b c-a d) f}+\frac {\sqrt {c+d} \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{4 b^2 \sqrt {a+b} d f}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}+\frac {(a+b)^{3/2} (5 b c-a d+2 b d) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 b^2 \sqrt {c+d} f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [A]
time = 2.25, antiderivative size = 784, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2900, 3126,
3140, 3132, 2890, 3077, 2897, 3075} \begin {gather*} \frac {\sqrt {c+d} \left (-a^2 d^2+6 a b c d+b^2 \left (3 c^2+4 d^2\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))}} \Pi \left (\frac {b (c+d)}{(a+b) d};\text {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 b^2 d f \sqrt {a+b}}+\frac {(a+b)^{3/2} (-a d+5 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))}} F\left (\text {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 b^2 f \sqrt {c+d}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} (a d+5 b c) \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 (\text {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 b f (b c-a d)}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(a d+5 b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2890
Rule 2897
Rule 2900
Rule 3075
Rule 3077
Rule 3126
Rule 3132
Rule 3140
Rubi steps
\begin {align*} \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} d \left (4 a^2 c-b^2 c+3 a b d\right )+d \left (3 a b c+2 a^2 d+b^2 d\right ) \sin (e+f x)+\frac {3}{2} b d (b c+a d) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^{3/2}} \, dx}{2 d}\\ &=\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b \left (a^2-b^2\right ) d \left (4 a c^2+b c d+a d^2\right )-\frac {1}{2} b \left (a^2-b^2\right ) d \left (2 b c^2+3 a c d+b d^2\right ) \sin (e+f x)-\frac {1}{4} b \left (a^2-b^2\right ) d^2 (5 b c+a d) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) d}\\ &=\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b \left (a^2-b^2\right ) d^2 \left (8 a^2 c^2-5 b^2 c^2+6 a b c d+3 a^2 d^2\right )-\frac {1}{2} b \left (a^2-b^2\right ) d^2 \left (5 a^2 c d+b^2 c d+3 a b \left (c^2+d^2\right )\right ) \sin (e+f x)-\frac {1}{4} b \left (a^2-b^2\right ) d^2 \left (6 a b c d-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}{2 b \left (a^2-b^2\right ) d^2}\\ &=\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {\int \frac {-\frac {1}{4} b^3 \left (a^2-b^2\right ) d^2 \left (8 a^2 c^2-5 b^2 c^2+6 a b c d+3 a^2 d^2\right )+\frac {1}{4} a^2 b \left (a^2-b^2\right ) d^2 \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right )+b \left (-\frac {1}{2} b^2 \left (a^2-b^2\right ) d^2 \left (5 a^2 c d+b^2 c d+3 a b \left (c^2+d^2\right )\right )+\frac {1}{2} a b \left (a^2-b^2\right ) d^2 \left (6 a b c d-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}{2 b^3 \left (a^2-b^2\right ) d^2}+\frac {\left (6 a b c d-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 b^2}\\ &=\frac {\sqrt {c+d} \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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 b^2 \sqrt {a+b} d f}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {((a+b) (b c-a d) (5 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 b}+\frac {((a+b) (b c-a d) (5 b c-a d+2 b d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^2}\\ &=\frac {\sqrt {a+b} (c-d) \sqrt {c+d} (5 b c+a d) E\left (\sin ^{-1}\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 b (b c-a d) f}+\frac {\sqrt {c+d} \left (6 a b c d-a^2 d^2+b^2 \left (3 c^2+4 d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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 b^2 \sqrt {a+b} d f}+\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \sqrt {a+b \sin (e+f x)}}-\frac {(5 b c+a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {a+b \sin (e+f x)}}+\frac {(a+b)^{3/2} (5 b c-a d+2 b d) F\left (\sin ^{-1}\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 b^2 \sqrt {c+d} f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1879\) vs. \(2(784)=1568\).
time = 9.64, size = 1879, normalized size = 2.40 \begin {gather*} -\frac {d \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}+\frac {-\frac {4 (-b c+a d) \left (8 a c^2+7 b c d+3 a d^2\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+a d) \left (8 b c^2+12 a c d+4 b d^2\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+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}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 \left (-5 b c d-a d^2\right ) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {\frac {a-b}{a+b}} (a+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a-b}{a+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\right )|\frac {2 (-b c+a d)}{(a-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{a+b \sin (e+f x)}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \sqrt {\frac {(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}-\frac {2 (-b c+a d) \left (\frac {((a+b) c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{8 f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 15.30, size = 278217, normalized size = 354.87
method | result | size |
default | \(\text {Expression too large to display}\) | \(278217\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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