Optimal. Leaf size=241 \[ -\frac {a^{5/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f} \]
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Rubi [A]
time = 0.37, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 3060,
2849, 2854, 211} \begin {gather*} -\frac {a^{5/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2842
Rule 2849
Rule 2854
Rule 3060
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+9 d)-\frac {1}{2} a^2 (3 c-13 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx}{3 d}\\ &=\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\left (a^2 \left (c^2-6 c d+25 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{8 d^2}\\ &=-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\left (a^2 (c+d) \left (c^2-6 c d+25 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 d^2}\\ &=-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}-\frac {\left (a^3 (c+d) \left (c^2-6 c d+25 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^2 f}\\ &=-\frac {a^{5/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 285, normalized size = 1.18 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{5/2} \left (\frac {(c+d) \left (c^2-6 c d+25 d^2\right ) \left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{5/2}}+\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (3 c^2-16 c d-79 d^2+4 d^2 \cos (2 (e+f x))-2 d (c+17 d) \sin (e+f x)\right )}{3 d^2}\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c +d \sin \left (f x +e \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs.
\(2 (221) = 442\).
time = 0.82, size = 1509, normalized size = 6.26 \begin {gather*} \left [\frac {3 \, {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3} + {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) + 8 \, {\left (8 \, a^{2} d^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} - 2 \, {\left (a^{2} c d + 13 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{2} - 16 \, a^{2} c d - 83 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a^{2} d^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} + 2 \, {\left (a^{2} c d + 17 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{192 \, {\left (d^{2} f \cos \left (f x + e\right ) + d^{2} f \sin \left (f x + e\right ) + d^{2} f\right )}}, \frac {3 \, {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3} + {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) + 4 \, {\left (8 \, a^{2} d^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} - 2 \, {\left (a^{2} c d + 13 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{2} - 16 \, a^{2} c d - 83 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a^{2} d^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} + 2 \, {\left (a^{2} c d + 17 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{96 \, {\left (d^{2} f \cos \left (f x + e\right ) + d^{2} f \sin \left (f x + e\right ) + d^{2} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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