\(\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx\) [223]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\text {Int}\left (\frac {1}{(d x)^{3/2} (a+b \arccos (c x))},x\right ) \]

[Out]

Unintegrable(1/(d*x)^(3/2)/(a+b*arccos(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx \]

[In]

Int[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])),x]

[Out]

Defer[Int][1/((d*x)^(3/2)*(a + b*ArcCos[c*x])), x]

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

Mathematica [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx \]

[In]

Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])),x]

[Out]

Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.94 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)/(b*d^2*x^2*arccos(c*x) + a*d^2*x^2), x)

Sympy [N/A]

Not integrable

Time = 3.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}\, dx \]

[In]

integrate(1/(d*x)**(3/2)/(a+b*acos(c*x)),x)

[Out]

Integral(1/((d*x)**(3/2)*(a + b*acos(c*x))), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((d*x)^(3/2)*(b*arccos(c*x) + a)), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x)),x, algorithm="giac")

[Out]

integrate(1/((d*x)^(3/2)*(b*arccos(c*x) + a)), x)

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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