Optimal. Leaf size=119 \[ \frac{3 A \sin (c+d x) (b \sec (c+d x))^{5/3} \text{Hypergeometric2F1}\left (-\frac{5}{6},\frac{1}{2},\frac{1}{6},\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \text{Hypergeometric2F1}\left (-\frac{4}{3},\frac{1}{2},-\frac{1}{3},\cos ^2(c+d x)\right )}{8 b^2 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0982565, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {16, 3787, 3772, 2643} \[ \frac{3 A \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \, _2F_1\left (-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx &=\frac{\int (b \sec (c+d x))^{8/3} (A+B \sec (c+d x)) \, dx}{b^2}\\ &=\frac{A \int (b \sec (c+d x))^{8/3} \, dx}{b^2}+\frac{B \int (b \sec (c+d x))^{11/3} \, dx}{b^3}\\ &=\frac{\left (A \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{8/3}} \, dx}{b^2}+\frac{\left (B \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{11/3}} \, dx}{b^3}\\ &=\frac{3 A \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \, _2F_1\left (-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.256236, size = 90, normalized size = 0.76 \[ -\frac{3 \left (-\tan ^2(c+d x)\right )^{3/2} \csc ^3(c+d x) (b \sec (c+d x))^{2/3} \left (11 A \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{7}{3},\sec ^2(c+d x)\right )+8 B \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{11}{6},\frac{17}{6},\sec ^2(c+d x)\right )\right )}{88 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{2} \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\sec \left ( dx+c \right ) \right ) \, dx \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 (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{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 \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )^{2}\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{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{\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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