Optimal. Leaf size=457 \[ -\frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (6 a^2 b (6 A+7 C)-75 a^3 B-24 a b^2 B+16 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{315 a^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (a^2 b (13 A+21 C)+75 a^3 B-12 a b^2 B+8 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left (6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-24 a b^3 B+16 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 1.64362, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (a^2 b (13 A+21 C)+75 a^3 B-12 a b^2 B+8 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (6 a^2 b (6 A+7 C)-75 a^3 B-24 a b^2 B+16 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left (6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-24 a b^3 B+16 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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{\sqrt{a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{\frac{1}{2} (A b+9 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac{3}{2} b (2 A+3 C) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{4 \int \frac{\frac{1}{4} \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right )-\frac{1}{4} a (47 A b+45 a B+63 b C) \sec (c+d x)-b (A b+9 a B) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{63 a}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 \int \frac{\frac{3}{8} \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right )+\frac{1}{8} a \left (2 A b^2+207 a b B+21 a^2 (7 A+9 C)\right ) \sec (c+d x)-\frac{1}{4} b \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{315 a^2}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{16 \int \frac{\frac{3}{16} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right )+\frac{3}{16} a \left (4 A b^3-75 a^3 B-6 a b^2 B-3 a^2 b (37 A+49 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{945 a^3}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right )\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 a^4}-\frac{\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{315 a^4}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 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}{315 a^4 \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{315 a^4 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 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}{315 a^4 \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 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}{315 a^4 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=-\frac{2 \left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 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)}}{315 a^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{315 a^4 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (A b+9 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.97829, size = 5993, normalized size = 13.11 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.9, size = 6551, normalized size = 14.3 \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] 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 )} \sqrt{b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \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 \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{9}{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 )} \sqrt{b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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