\(\int \frac {(d+e x^2)^{3/2} (a+b \csc ^{-1}(c x))}{x} \, dx\) [130]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.92 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{x}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 72.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]

[In]

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

[Out]

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