Optimal. Leaf size=150 \[ \frac{2 B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.154279, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4047, 3769, 3771, 2641, 4045, 2639} \[ \frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3769
Rule 3771
Rule 2641
Rule 4045
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac{B \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{b}+\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}+\frac{B \int \sqrt{b \sec (c+d x)} \, dx}{3 b^3}+\frac{(3 A+5 C) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx}{5 b^2}\\ &=\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}+\frac{(3 A+5 C) \int \sqrt{\cos (c+d x)} \, dx}{5 b^2 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^3}\\ &=\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 A \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}\\ \end{align*}
Mathematica [C] time = 1.97786, size = 169, normalized size = 1.13 \[ \frac{e^{-i d x} (\cos (d x)+i \sin (d x)) \sqrt{b \sec (c+d x)} \left (-2 i (3 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+10 B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+\cos (c+d x) (3 A \sin (2 (c+d x))+6 i (3 A+5 C)+10 B \sin (c+d x))\right )}{15 b^3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.229, size = 766, normalized size = 5.1 \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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{\left (b \sec \left (d x + c\right )\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 (\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 )}}{b^{3} \sec \left (d x + c\right )^{3}}, 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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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