3.1.68 \(\int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx\) [68]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right ) \]

[Out]

Unintegrable((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x)

________________________________________________________________________________________

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 = {} \begin {gather*} \int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)),x]

[Out]

Defer[Int][(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)), x]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx &=\int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 15.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)),x]

[Out]

Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)), x]

________________________________________________________________________________________

Maple [A]
time = 5.21, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x)

[Out]

int((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

1/2*(2*(b*d^2*x^2*e + b*d^3*x)*sqrt(d)*integrate(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/((x^3*e + d*x^2)*sqrt
(x*e + d)), x) - 2*(3*a*x*e + a*d)*sqrt(x*e + d)*sqrt(d) - 3*(a*x^2*e^2 + a*d*x*e)*log(x*e/(x*e + 2*sqrt(x*e +
d)*sqrt(d) + 2*d)))/((d^2*x^2*e + d^3*x)*sqrt(d))

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

integral((b*arccsc(c*x) + a)*sqrt(x*e + d)/(x^4*e^2 + 2*d*x^3*e + d^2*x^2), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))/x**2/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*acsc(c*x))/(x**2*(d + e*x)**(3/2)), x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/(x^2*(d + e*x)^(3/2)),x)

[Out]

int((a + b*asin(1/(c*x)))/(x^2*(d + e*x)^(3/2)), x)

________________________________________________________________________________________