Optimal. Leaf size=27 \[ \frac{b^2 x \sqrt{b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0026389, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 8} \[ \frac{b^2 x \sqrt{b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 8
Rubi steps
\begin{align*} \int \frac{(b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int 1 \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{b^2 x \sqrt{b \sec (c+d x)}}{\sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0194576, size = 24, normalized size = 0.89 \[ \frac{x (b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 32, normalized size = 1.2 \begin{align*}{\frac{dx+c}{d} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82461, size = 35, normalized size = 1.3 \begin{align*} \frac{2 \, b^{\frac{5}{2}} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97332, size = 271, normalized size = 10.04 \begin{align*} \left [\frac{\sqrt{-b} b^{2} \log \left (-2 \, \sqrt{-b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{2 \, d}, \frac{b^{\frac{5}{2}} \arctan \left (\frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{b} \sqrt{\cos \left (d x + c\right )}}\right )}{d}\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{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}}}{\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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