3.3.16 \(\int \sqrt {d x} (a+b \text {ArcCos}(c x))^3 \, dx\) [216]

Optimal. Leaf size=67 \[ \frac {2 (d x)^{3/2} (a+b \text {ArcCos}(c x))^3}{3 d}+\frac {2 b c \text {Int}\left (\frac {(d x)^{3/2} (a+b \text {ArcCos}(c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{d} \]

[Out]

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

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Rubi [A]
time = 0.11, 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 = {} \begin {gather*} \int \sqrt {d x} (a+b \text {ArcCos}(c x))^3 \, dx \end {gather*}

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*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]
time = 0.17, size = 0, normalized size = 0.00 \[\int \left (a +b \arccos \left (c x \right )\right )^{3} \sqrt {d x}\, dx\]

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

2/3*b^3*sqrt(d)*x^(3/2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 1/6*a^3*c^2*sqrt(d)*(4*x^(3/2)/c^2 + 6*
arctan(sqrt(c)*sqrt(x))/c^(7/2) + 3*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(7/2)) + 3*a*b^2*c^2*sq
rt(d)*integrate(x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*x^2 - 1), x) + 3*a^2*b*c^2*sqrt(d)*i
ntegrate(x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x) - 2*b^3*c*sqrt(d)*integrate(sqrt
(c*x + 1)*sqrt(-c*x + 1)*x^(3/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*x^2 - 1), x) - 1/2*a^3*sqrt
(d)*(2*arctan(sqrt(c)*sqrt(x))/c^(3/2) + log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(3/2)) - 3*a*b^2*s
qrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*x^2 - 1), x) - 3*a^2*b*sqrt(d)*inte
grate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}\, dx \end {gather*}

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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3\,\sqrt {d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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