3.216 \(\int \sqrt{d x} (a+b \cos ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=66 \[ \frac{2 b c \text{Unintegrable}\left (\frac{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}},x\right )}{d}+\frac{2 (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.166553, 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 = {} \[ \int \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 135.147, size = 0, normalized size = 0. \[ \int \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.349, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( cx \right ) \right ) ^{3}\sqrt{dx}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arccos(c*x))^3*(d*x)^(1/2),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((a+b*arccos(c*x))^3*(d*x)^(1/2),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} \arccos \left (c x\right )^{3} + 3 \, a b^{2} \arccos \left (c x\right )^{2} + 3 \, a^{2} b \arccos \left (c x\right ) + a^{3}\right )} \sqrt{d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d x}{\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((a+b*arccos(c*x))^3*(d*x)^(1/2),x, algorithm="giac")

[Out]

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