Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}},x\right ) \]
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Rubi [A] time = 0.02, 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 = {} \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx &=\int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 23.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \sqrt {x} \sec \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \sqrt {x} \sec \left (d \sqrt {x} + c\right ) + a^{2} \sqrt {x}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}{x^{\frac {3}{2}}}\, dx \]
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 \[ \frac {4 \, b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + 4 \, {\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} + d \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} + d \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x}\right )} x - 2 \, {\left (a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} d\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} 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 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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