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

Optimal. Leaf size=68 \[ \frac{6 b c \text{Unintegrable}\left (\frac{(d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}},x\right )}{5 d}+\frac{2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^3}{5 d} \]

[Out]

(2*(d*x)^(5/2)*(a + b*ArcCos[c*x])^3)/(5*d) + (6*b*c*Unintegrable[((d*x)^(5/2)*(a + b*ArcCos[c*x])^2)/Sqrt[1 -
c^2*x^2], x])/(5*d)

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Rubi [A]  time = 0.164384, 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 = {} \[ \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*(d*x)^(5/2)*(a + b*ArcCos[c*x])^3)/(5*d) + (6*b*c*Defer[Int][((d*x)^(5/2)*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c
^2*x^2], x])/(5*d)

Rubi steps

\begin{align*} \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx &=\frac{2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^3}{5 d}+\frac{(6 b c) \int \frac{(d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{5 d}\\ \end{align*}

Mathematica [A]  time = 30.752, size = 0, normalized size = 0. \[ \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{{\frac{3}{2}}} \left ( a+b\arccos \left ( cx \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} d x \arccos \left (c x\right )^{3} + 3 \, a b^{2} d x \arccos \left (c x\right )^{2} + 3 \, a^{2} b d x \arccos \left (c x\right ) + a^{3} d x\right )} \sqrt{d x}, 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))^3,x, algorithm="fricas")

[Out]

integral((b^3*d*x*arccos(c*x)^3 + 3*a*b^2*d*x*arccos(c*x)^2 + 3*a^2*b*d*x*arccos(c*x) + a^3*d*x)*sqrt(d*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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