3.55 \(\int \frac{a+b \csc (c+d \sqrt{x})}{x^{5/2}} \, dx\)

Optimal. Leaf size=31 \[ b \text{Unintegrable}\left (\frac{\csc \left (c+d \sqrt{x}\right )}{x^{5/2}},x\right )-\frac{2 a}{3 x^{3/2}} \]

[Out]

(-2*a)/(3*x^(3/2)) + b*Unintegrable[Csc[c + d*Sqrt[x]]/x^(5/2), x]

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Rubi [A]  time = 0.0140257, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \csc \left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*a)/(3*x^(3/2)) + b*Defer[Int][Csc[c + d*Sqrt[x]]/x^(5/2), x]

Rubi steps

\begin{align*} \int \frac{a+b \csc \left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx &=\int \left (\frac{a}{x^{5/2}}+\frac{b \csc \left (c+d \sqrt{x}\right )}{x^{5/2}}\right ) \, dx\\ &=-\frac{2 a}{3 x^{3/2}}+b \int \frac{\csc \left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 6.6027, size = 0, normalized size = 0. \[ \int \frac{a+b \csc \left (c+d \sqrt{x}\right )}{x^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.131, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ){x}^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [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.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \sqrt{x} \csc \left (d \sqrt{x} + c\right ) + a \sqrt{x}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \csc{\left (c + d \sqrt{x} \right )}}{x^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \csc \left (d \sqrt{x} + c\right ) + a}{x^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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