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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

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

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Mathematica [A]  time = 3.43, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b d^{2} x^{2} \arccos \left (c x\right ) + a d^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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