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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 6.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 2.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (3 \, \sqrt {d} e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right ) - 3 \, \sqrt {e x^{2} + d} e - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a + {\left (e \int \frac {\sqrt {e x^{2} + d} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} + d \int \frac {\sqrt {e x^{2} + d} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\,{d x}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*sqrt(d)*e*arcsinh(d/(sqrt(d*e)*abs(x))) - 3*sqrt(e*x^2 + d)*e - (e*x^2 + d)^(3/2)*e/d + (e*x^2 + d)^(5
/2)/(d*x^2))*a + (e*integrate(sqrt(e*x^2 + d)*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/x, x) + d*integrate(sqrt
(e*x^2 + d)*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/x^3, x))*b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________