Optimal. Leaf size=218 \[ \frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{7/2} f \sqrt {c+d}}+\frac {2 a^3 \left (5 A d (3 c-7 d)-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (-5 A d+5 B c-8 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 d f} \]
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Rubi [A] time = 0.88, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2976, 2981, 2773, 208} \[ \frac {2 a^3 \left (5 A d (3 c-7 d)-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (-5 A d+5 B c-8 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 d^2 f}+\frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{7/2} f \sqrt {c+d}}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 d f} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2773
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx &=-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {2 \int \frac {(a+a \sin (e+f x))^{3/2} \left (\frac {1}{2} a (3 B c+5 A d)-\frac {1}{2} a (5 B c-5 A d-8 B d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{5 d}\\ &=\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {4 \int \frac {\sqrt {a+a \sin (e+f x)} \left (-\frac {1}{4} a^2 (B c (5 c-17 d)-5 A d (c+3 d))-\frac {1}{4} a^2 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{15 d^2}\\ &=\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}-\frac {\left (a^2 (c-d)^2 (B c-A d)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{d^3}\\ &=\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {\left (2 a^3 (c-d)^2 (B c-A d)\right ) \operatorname {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d^3 f}\\ &=\frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{7/2} \sqrt {c+d} f}+\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}\\ \end {align*}
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Mathematica [B] time = 5.82, size = 450, normalized size = 2.06 \[ \frac {(a (\sin (e+f x)+1))^{5/2} \left (30 \sqrt {d} \left (A d (5 d-2 c)+B \left (2 c^2-5 c d+5 d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )-30 \sqrt {d} \left (A d (5 d-2 c)+B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-5 d^{3/2} (2 A d-2 B c+5 B d) \sin \left (\frac {3}{2} (e+f x)\right )-5 d^{3/2} (2 A d-2 B c+5 B d) \cos \left (\frac {3}{2} (e+f x)\right )-\frac {15 (c-d)^2 (B c-A d) \left (2 \log \left (\sqrt {d} \sqrt {c+d} \left (\tan ^2\left (\frac {1}{4} (e+f x)\right )+2 \tan \left (\frac {1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{\sqrt {c+d}}+\frac {15 (c-d)^2 (B c-A d) \left (2 \log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+c+d\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{\sqrt {c+d}}-3 B d^{5/2} \sin \left (\frac {5}{2} (e+f x)\right )+3 B d^{5/2} \cos \left (\frac {5}{2} (e+f x)\right )\right )}{30 d^{7/2} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 2.27, size = 1314, normalized size = 6.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.05, size = 543, normalized size = 2.49 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-3 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a \left (c +d \right ) d}\, d^{2}+5 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a \,d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} c^{2} d +30 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} c \,d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} d^{3}-5 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a c d +20 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a \,d^{2}+15 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} c^{3}-30 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} c^{2} d +15 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{3} c \,d^{2}+15 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c d -45 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} d^{2}-15 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c^{2}+45 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c d -60 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} d^{2}\right )}{15 d^{3} \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{d \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{c+d\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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