Optimal. Leaf size=171 \[ \frac {2 A b^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 A b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}-\frac {6 b^3 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {6 b^2 B \sin (c+d x) \sqrt {b \sec (c+d x)}}{5 d}+\frac {2 B \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3787, 3768, 3771, 2641, 2639} \[ \frac {2 A b^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 A b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac {6 b^2 B \sin (c+d x) \sqrt {b \sec (c+d x)}}{5 d}-\frac {6 b^3 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 B \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rubi steps
\begin {align*} \int (b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=A \int (b \sec (c+d x))^{5/2} \, dx+\frac {B \int (b \sec (c+d x))^{7/2} \, dx}{b}\\ &=\frac {2 A b (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {2 B (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{3} \left (A b^2\right ) \int \sqrt {b \sec (c+d x)} \, dx+\frac {1}{5} (3 b B) \int (b \sec (c+d x))^{3/2} \, dx\\ &=\frac {6 b^2 B \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 A b (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {2 B (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {1}{5} \left (3 b^3 B\right ) \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx+\frac {1}{3} \left (A b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 A b^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {6 b^2 B \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 A b (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {2 B (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {\left (3 b^3 B\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=-\frac {6 b^3 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 A b^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {6 b^2 B \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 A b (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {2 B (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 99, normalized size = 0.58 \[ \frac {(b \sec (c+d x))^{5/2} \left (10 A \sin (2 (c+d x))+20 A \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+21 B \sin (c+d x)+9 B \sin (3 (c+d x))-36 B \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b^{2} \sec \left (d x + c\right )^{3} + A b^{2} \sec \left (d x + c\right )^{2}\right )} \sqrt {b \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.03, size = 518, normalized size = 3.03 \[ \frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (5 i A \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-9 i B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+9 i B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+5 i A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-9 i B \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+9 i B \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-5 A \left (\cos ^{3}\left (d x +c \right )\right )-9 B \left (\cos ^{3}\left (d x +c \right )\right )+6 B \left (\cos ^{2}\left (d x +c \right )\right )+5 A \cos \left (d x +c \right )+3 B \right ) \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}}{15 d \sin \left (d x +c \right )^{5}} \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (A + B \sec {\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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