Optimal. Leaf size=610 \[ \frac {2 \tan (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \tan (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \tan (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{3465 b^2 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-40 a^4 C+10 a^3 b (11 B-3 C)-15 a^2 b^2 (33 A-121 B+19 C)+6 a b^3 (660 A-209 B+505 C)-3 b^4 (275 A-539 B+225 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3465 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3465 b^4 d}+\frac {2 (11 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d} \]
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Rubi [A] time = 2.11, antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4092, 4082, 4002, 4005, 3832, 4004} \[ \frac {2 \tan (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \tan (c+d x) \left (110 a^2 b B-40 a^3 C-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \tan (c+d x) \left (-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 C-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{3465 b^2 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-15 a^2 b^2 (33 A-121 B+19 C)+10 a^3 b (11 B-3 C)-40 a^4 C+6 a b^3 (660 A-209 B+505 C)-3 b^4 (275 A-539 B+225 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3465 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B+110 a^4 b B-40 a^5 C-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3465 b^4 d}+\frac {2 (11 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d} \]
Antiderivative was successfully verified.
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Rule 3832
Rule 4002
Rule 4004
Rule 4005
Rule 4082
Rule 4092
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {2 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (a C+\frac {1}{2} b (11 A+9 C) \sec (c+d x)+\frac {1}{2} (11 b B-4 a C) \sec ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {1}{4} b (77 b B-10 a C)+\frac {1}{4} \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) \sec (c+d x)\right ) \, dx}{99 b^2}\\ &=\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {8 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {3}{8} b \left (165 A b^2+143 a b B-10 a^2 C+135 b^2 C\right )-\frac {1}{8} \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) \sec (c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {16 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{16} b \left (605 a^2 b B+539 b^3 B-10 a^3 C+10 a b^2 (132 A+101 C)\right )-\frac {3}{16} \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sec (c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {32 \int \frac {\sec (c+d x) \left (\frac {3}{32} b \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+75 b^4 (11 A+9 C)+15 a^2 b^2 (297 A+221 C)\right )-\frac {3}{32} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}-\frac {\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 b^2}+\frac {\left (32 \left (\frac {3}{32} b \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+75 b^4 (11 A+9 C)+15 a^2 b^2 (297 A+221 C)\right )+\frac {3}{32} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right )\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{10395 b^2}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (a^3 b (110 B-30 C)-40 a^4 C-15 a^2 b^2 (33 A-121 B+19 C)-3 b^4 (275 A-539 B+225 C)+6 a b^3 (660 A-209 B+505 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^3 d}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}\\ \end {align*}
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Mathematica [A] time = 21.77, size = 1090, normalized size = 1.79 \[ \frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac {4}{99} \left (11 B \sin (c+d x) b^2+23 a C \sin (c+d x) b\right ) \sec ^4(c+d x)+\frac {4}{11} b^2 C \tan (c+d x) \sec ^4(c+d x)+\frac {4}{693} \left (113 C \sin (c+d x) a^2+209 b B \sin (c+d x) a+99 A b^2 \sin (c+d x)+81 b^2 C \sin (c+d x)\right ) \sec ^3(c+d x)+\frac {4 \left (15 C \sin (c+d x) a^3+825 b B \sin (c+d x) a^2+1485 A b^2 \sin (c+d x) a+1145 b^2 C \sin (c+d x) a+539 b^3 B \sin (c+d x)\right ) \sec ^2(c+d x)}{3465 b}+\frac {4 \left (-20 C \sin (c+d x) a^4+55 b B \sin (c+d x) a^3+1485 A b^2 \sin (c+d x) a^2+1025 b^2 C \sin (c+d x) a^2+1793 b^3 B \sin (c+d x) a+825 A b^4 \sin (c+d x)+675 b^4 C \sin (c+d x)\right ) \sec (c+d x)}{3465 b^2}+\frac {4 \left (40 C a^5-110 b B a^4+495 A b^2 a^3+255 b^2 C a^3+3069 b^3 B a^2+4785 A b^4 a+3705 b^4 C a+1617 b^5 B\right ) \sin (c+d x)}{3465 b^3}\right )}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac {4 (a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left ((a+b) \left (40 C a^5-110 b B a^4+15 b^2 (33 A+17 C) a^3+3069 b^3 B a^2+15 b^4 (319 A+247 C) a+1617 b^5 B\right ) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )-b (a+b) \left (40 C a^4-10 b (11 B+3 C) a^3+15 b^2 (33 A+121 B+19 C) a^2+6 b^3 (660 A+209 B+505 C) a+3 b^4 (275 A+539 B+225 C)\right ) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )+\left (40 C a^5-110 b B a^4+15 b^2 (33 A+17 C) a^3+3069 b^3 B a^2+15 b^4 (319 A+247 C) a+1617 b^5 B\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (-b \tan ^4\left (\frac {1}{2} (c+d x)\right )+a \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )-1\right )^2+b\right )\right )}{3465 b^3 d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt {\frac {-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \sec \left (d x + c\right )^{6} + {\left (2 \, C a b + B b^{2}\right )} \sec \left (d x + c\right )^{5} + A a^{2} \sec \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{4} + {\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )^{3}\right )} \sqrt {b \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.13, size = 7208, normalized size = 11.82 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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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