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

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)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

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*}

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Mathematica [A]  time = 14.91, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

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]

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fricas [A]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \]

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

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

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)

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maple [F(-2)]  time = 180.00, 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)

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

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

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

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)

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

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)

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