Optimal. Leaf size=62 \[ \frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2641} \[ \frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \sec ^{\frac {5}{2}}(a+b x) \, dx &=\frac {2 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{3 b}+\frac {1}{3} \int \sqrt {\sec (a+b x)} \, dx\\ &=\frac {2 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{3 b}+\frac {1}{3} \left (\sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{3 b}+\frac {2 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 46, normalized size = 0.74 \[ \frac {2 \sec ^{\frac {3}{2}}(a+b x) \left (\sin (a+b x)+\cos ^{\frac {3}{2}}(a+b x) F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sec \left (b x + a\right )^{\frac {5}{2}}, x\right ) \]
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 \[ \int \sec \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.07, size = 213, normalized size = 3.44 \[ -\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) b} \]
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 \[ \int \sec \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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 \[ \int \sec ^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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