3.66 \(\int \frac {1}{x^{5/2} (a+b \text {sech}(c+d \sqrt {x}))} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x^{5/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )},x\right ) \]

[Out]

Unintegrable(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{x^{5/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])),x]

[Out]

Defer[Int][1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])), x]

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \, dx &=\int \frac {1}{x^{5/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \, dx\\ \end {align*}

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Mathematica [A]  time = 9.03, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])),x]

[Out]

Integrate[1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x}}{b x^{3} \operatorname {sech}\left (d \sqrt {x} + c\right ) + a x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

integral(sqrt(x)/(b*x^3*sech(d*sqrt(x) + c) + a*x^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} x^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x, algorithm="giac")

[Out]

integrate(1/((b*sech(d*sqrt(x) + c) + a)*x^(5/2)), x)

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maple [A]  time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x)

[Out]

int(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, b \int \frac {e^{\left (d \sqrt {x} + c\right )}}{a^{2} x^{\frac {5}{2}} e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + 2 \, a b x^{\frac {5}{2}} e^{\left (d \sqrt {x} + c\right )} + a^{2} x^{\frac {5}{2}}}\,{d x} - \frac {2}{3 \, a x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(a+b*sech(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

-2*b*integrate(e^(d*sqrt(x) + c)/(a^2*x^(5/2)*e^(2*d*sqrt(x) + 2*c) + 2*a*b*x^(5/2)*e^(d*sqrt(x) + c) + a^2*x^
(5/2)), x) - 2/3/(a*x^(3/2))

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^{5/2}\,\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b/cosh(c + d*x^(1/2)))),x)

[Out]

int(1/(x^(5/2)*(a + b/cosh(c + d*x^(1/2)))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(a+b*sech(c+d*x**(1/2))),x)

[Out]

Integral(1/(x**(5/2)*(a + b*sech(c + d*sqrt(x)))), x)

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