Optimal. Leaf size=164 \[ \frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.25, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2862, 2669, 2642, 2641} \[ \frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2691
Rule 2862
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}-\frac {2 \int \frac {(a+b \sin (c+d x)) \left (-\frac {a^2}{2}+2 b^2+\frac {3}{2} a b \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}-\frac {4 \int \frac {-\frac {3}{4} a \left (a^2-6 b^2\right )+\frac {3}{4} b \left (a^2+4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{9 e^2}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 b \left (a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 a \left (a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 103, normalized size = 0.63 \[ \frac {2 a^3 \sin (c+d x)+2 a \left (a^2-6 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 a^2 b+6 a b^2 \sin (c+d x)+3 b^3 \cos (2 (c+d x))+5 b^3}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.60, size = 384, normalized size = 2.34 \[ -\frac {2 \left (2 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a^{3} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a \,b^{2} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}+2 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+4 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d} \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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