Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{3}}(c+d x)},x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{3}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{3}}(c+d x)} \, dx &=\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{3}}(c+d x)} \, dx\\ \end {align*}
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Mathematica [A] time = 36.20, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{3}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {5}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{\sec \left (d x +c \right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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