Optimal. Leaf size=565 \[ \frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-110 a b^3 B+40 A b^4\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left (5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{231 d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left (15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
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Rubi [A] time = 2.29866, antiderivative size = 565, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \sin (c+d x) \left (5 a^2 b (229 A+297 C)+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{231 d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left (-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-55 a b^3 B+20 A b^4\right ) \sqrt{a+b \sec (c+d x)}}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-110 a b^3 B+40 A b^4\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4104
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2}{11} \int \frac{(a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+11 a B)+\frac{1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)+\frac{1}{2} b (4 A+11 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4}{99} \int \frac{\sqrt{a+b \sec (c+d x)} \left (\frac{3}{4} \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right )+\frac{1}{4} \left (152 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \sec (c+d x)+\frac{1}{4} b (56 A b+44 a B+99 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{8}{693} \int \frac{\frac{1}{8} \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right )+\frac{1}{8} \left (1507 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1531 A+2079 C)\right ) \sec (c+d x)+\frac{1}{8} b \left (836 a b B+36 a^2 (9 A+11 C)+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{16 \int \frac{\frac{3}{16} \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right )-\frac{1}{16} a \left (1617 a^3 B+6655 a b^2 B+15 a^2 b (337 A+429 C)+5 b^3 (461 A+693 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{3465 a}\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{32 \int \frac{\frac{3}{32} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right )+\frac{3}{32} a \left (10 A b^4+2871 a^3 b B+1705 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (221 A+297 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{10395 a^2}\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right )\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{3465 a^3}+\frac{\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{3465 a^3}\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{3465 a^3 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{3465 a^3 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{3465 a^3 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{3465 a^3 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{2 \left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{3465 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{3465 a^3 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 7.3617, size = 7479, normalized size = 13.24 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 1.17, size = 7971, normalized size = 14.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \sec \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{11}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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 )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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