Optimal. Leaf size=136 \[ \frac{2 b B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 d}-\frac{2 A b^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.10093, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3787, 3768, 3771, 2639, 2641} \[ -\frac{2 A b^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3787
Rule 3768
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=A \int (b \sec (c+d x))^{3/2} \, dx+\frac{B \int (b \sec (c+d x))^{5/2} \, dx}{b}\\ &=\frac{2 A b \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}-\left (A b^2\right ) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx+\frac{1}{3} (b B) \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 A b \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac{\left (A b^2\right ) \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{1}{3} \left (b B \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 A b^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b 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 d}+\frac{2 A b \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.274528, size = 87, normalized size = 0.64 \[ \frac{(b \sec (c+d x))^{3/2} \left (2 B \cos ^{\frac{3}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+2 \sin (c+d x) (3 A \cos (c+d x)+B)-6 A \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.252, size = 500, normalized size = 3.7 \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 (b \sec \left (d x + c\right )\right )^{\frac{3}{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 b \sec \left (d x + c\right )^{2} + A b \sec \left (d x + c\right )\right )} \sqrt{b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}} \left (A + B \sec{\left (c + d x \right )}\right )\, dx \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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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