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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0282532, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.47324, size = 0, normalized size = 0. \[ \int \frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\arccos \left ( cx \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{b \arccos \left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x} d x}{b \arccos \left (c x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{a + b \operatorname{acos}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{b \arccos \left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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