Optimal. Leaf size=199 \[ \frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{6 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.179355, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3997, 3787, 3768, 3771, 2641, 2639} \[ \frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{6 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{6 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3997
Rule 3787
Rule 3768
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sec ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (7 A+5 B)+\frac{7}{2} a (A+B) \sec (c+d x)\right ) \, dx\\ &=\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+(a (A+B)) \int \sec ^{\frac{7}{2}}(c+d x) \, dx+\frac{1}{7} (a (7 A+5 B)) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 a (7 A+5 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (A+B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{5} (3 a (A+B)) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{21} (a (7 A+5 B)) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{6 a (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (A+B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{5} (3 a (A+B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (a (7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{6 a (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (A+B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{5} \left (3 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{6 a (A+B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (7 A+5 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{6 a (A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (A+B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a B \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.739832, size = 200, normalized size = 1.01 \[ \frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1) (A+B \sec (c+d x)) \left (5 (7 A+5 B) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-63 (A+B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+63 A \sin (c+d x)+35 A \tan (c+d x)+21 A \tan (c+d x) \sec (c+d x)+63 B \sin (c+d x)+25 B \tan (c+d x)+15 B \tan (c+d x) \sec ^2(c+d x)+21 B \tan (c+d x) \sec (c+d x)\right )}{105 d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.554, size = 691, normalized size = 3.5 \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{\left (B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{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 ({\left (B a \sec \left (d x + c\right )^{4} +{\left (A + B\right )} a \sec \left (d x + c\right )^{3} + A a \sec \left (d x + c\right )^{2}\right )} \sqrt{\sec \left (d x + c\right )}, 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{\left (B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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