Optimal. Leaf size=210 \[ \frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 b d}+\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d} \]
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Rubi [A] time = 0.71332, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac{2 a (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b d} \]
Antiderivative was successfully verified.
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Rule 4033
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx &=\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}+\frac{2 \int \frac{\sqrt{\sec (c+d x)} \left (\frac{a B}{2}+\frac{1}{2} b B \sec (c+d x)+\frac{3}{2} (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b}\\ &=\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^2 d}+\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}+\frac{4 \int \frac{-\frac{3}{4} a (A b-a B)-\frac{1}{4} b (3 A b-4 a B) \sec (c+d x)-\frac{1}{4} \left (3 a A b-3 a^2 B-b^2 B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^2}\\ &=\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^2 d}+\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}+\frac{4 \int \frac{-\frac{3}{4} a^2 (A b-a B)-\left (\frac{1}{4} a b (3 A b-4 a B)-\frac{3}{4} a b (A b-a B)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^2}-\frac{(a (A b-a B)) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{b^2}\\ &=\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^2 d}+\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}+\frac{B \int \sqrt{\sec (c+d x)} \, dx}{3 b}-\frac{(A b-a B) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{b^2}-\frac{\left (a (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^2}\\ &=-\frac{2 a (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b^2 (a+b) d}+\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^2 d}+\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}+\frac{\left (B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b}-\frac{\left ((A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b^2}\\ &=-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b^2 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 b d}-\frac{2 a (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b^2 (a+b) d}+\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{b^2 d}+\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 3.65854, size = 229, normalized size = 1.09 \[ -\frac{\cot (c+d x) \left (-2 \left (3 a^2 B+3 a b (B-A)+b^2 (B-3 A)\right ) \sqrt{-\tan ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )-6 a^2 B \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-6 b (A b-a B) \sqrt{-\tan ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+6 a A b \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-b^2 B \sec ^{\frac{5}{2}}(c+d x)+b^2 B \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)\right )}{3 b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 5.356, size = 466, normalized size = 2.2 \begin{align*} \text{result 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 (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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