\(\int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx\) [187]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\text {Int}\left (\frac {(a+b \arccos (c x))^{5/2}}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*arccos(c*x))^(5/2)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx \]

[In]

Int[(a + b*ArcCos[c*x])^(5/2)/x^2,x]

[Out]

Defer[Int][(a + b*ArcCos[c*x])^(5/2)/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.91 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{\frac {5}{2}}}{x^{2}}d x\]

[In]

int((a+b*arccos(c*x))^(5/2)/x^2,x)

[Out]

int((a+b*arccos(c*x))^(5/2)/x^2,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 24.82 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}}{x^{2}}\, dx \]

[In]

integrate((a+b*acos(c*x))**(5/2)/x**2,x)

[Out]

Integral((a + b*acos(c*x))**(5/2)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate((b*arccos(c*x) + a)^(5/2)/x^2, x)

Giac [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate((b*arccos(c*x) + a)^(5/2)/x^2, x)

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^{5/2}}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2}}{x^2} \,d x \]

[In]

int((a + b*acos(c*x))^(5/2)/x^2,x)

[Out]

int((a + b*acos(c*x))^(5/2)/x^2, x)