Optimal. Leaf size=266 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (9 a B+9 A b+7 b C)}{15 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d} \]
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Rubi [A] time = 0.300583, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (9 a B+9 A b+7 b C)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a A+5 a C+5 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d} \]
Antiderivative was successfully verified.
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Rule 4076
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2}{9} \int \sec ^{\frac{5}{2}}(c+d x) \left (\frac{9 a A}{2}+\frac{1}{2} (9 A b+9 a B+7 b C) \sec (c+d x)+\frac{9}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2}{9} \int \sec ^{\frac{5}{2}}(c+d x) \left (\frac{9 a A}{2}+\frac{9}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx+\frac{1}{9} (9 A b+9 a B+7 b C) \int \sec ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 (9 A b+9 a B+7 b C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{7} (7 a A+5 b B+5 a C) \int \sec ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{15} (9 A b+9 a B+7 b C) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 (9 A b+9 a B+7 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (7 a A+5 b B+5 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 (9 A b+9 a B+7 b C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} (7 a A+5 b B+5 a C) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} (-9 A b-9 a B-7 b C) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 (9 A b+9 a B+7 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (7 a A+5 b B+5 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 (9 A b+9 a B+7 b C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} \left ((7 a A+5 b B+5 a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left ((-9 A b-9 a B-7 b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (9 A b+9 a B+7 b C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (7 a A+5 b B+5 a C) \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{2 (9 A b+9 a B+7 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (7 a A+5 b B+5 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 (9 A b+9 a B+7 b C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [C] time = 7.30943, size = 1262, normalized size = 4.74 \[ \frac{4 a A \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 b B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 a C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} A b e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{14 \sqrt{2} b C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{4 b C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{4 \sec (c) (7 b C \sin (c)+9 b B \sin (d x)+9 a C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac{4 \sec (c) (45 b B \sin (c)+45 a C \sin (c)+63 A b \sin (d x)+63 a B \sin (d x)+49 b C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{4 \sec (c) (63 A b \sin (c)+63 a B \sin (c)+49 b C \sin (c)+105 a A \sin (d x)+75 b B \sin (d x)+75 a C \sin (d x)) \sec (c+d x)}{315 d}+\frac{4 (9 A b+7 C b+9 a B) \cos (d x) \csc (c)}{15 d}+\frac{4 (7 a A+5 b B+5 a C) \tan (c)}{21 d}\right )}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 8.952, size = 1020, normalized size = 3.8 \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(-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 ({\left (C b \sec \left (d x + c\right )^{5} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{4} + A a \sec \left (d x + c\right )^{2} +{\left (B a + A b\right )} \sec \left (d x + c\right )^{3}\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 (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 )} \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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