3.3.15 \(\int (d x)^{3/2} (a+b \text {ArcCos}(c x))^3 \, dx\) [215]

Optimal. Leaf size=69 \[ \frac {2 (d x)^{5/2} (a+b \text {ArcCos}(c x))^3}{5 d}+\frac {6 b c \text {Int}\left (\frac {(d x)^{5/2} (a+b \text {ArcCos}(c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{5 d} \]

[Out]

2/5*(d*x)^(5/2)*(a+b*arccos(c*x))^3/d+6/5*b*c*Unintegrable((d*x)^(5/2)*(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 (d x)^{3/2} (a+b \text {ArcCos}(c x))^3 \, dx \end {gather*}

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*}

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Mathematica [A]
time = 113.65, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^{3/2} (a+b \text {ArcCos}(c x))^3 \, dx \end {gather*}

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 = 0.18, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{3}\, dx\]

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

[Out]

2/5*b^3*d^(3/2)*x^(5/2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 1/10*a^3*c^2*d^(3/2)*(4*(c^2*x^(5/2) +
5*sqrt(x))/c^4 - 10*arctan(sqrt(c)*sqrt(x))/c^(9/2) + 5*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(9/
2)) + 15*a*b^2*c^2*d^(3/2)*integrate(1/5*x^(7/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*x^2 - 1), x
) + 15*a^2*b*c^2*d^(3/2)*integrate(1/5*x^(7/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*x^2 - 1), x) -
6*b^3*c*d^(3/2)*integrate(1/5*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^
2/(c^2*x^2 - 1), x) - 1/2*a^3*d^(3/2)*(4*sqrt(x)/c^2 - 2*arctan(sqrt(c)*sqrt(x))/c^(5/2) + log((c*sqrt(x) - sq
rt(c))/(c*sqrt(x) + sqrt(c)))/c^(5/2)) - 15*a*b^2*d^(3/2)*integrate(1/5*x^(3/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x
+ 1)/(c*x))^2/(c^2*x^2 - 1), x) - 15*a^2*b*d^(3/2)*integrate(1/5*x^(3/2)*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((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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {3}{2}} \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((d*x)**(3/2)*(a+b*acos(c*x))**3,x)

[Out]

Integral((d*x)**(3/2)*(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((d*x)^(3/2)*(a+b*arccos(c*x))^3,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\,{\left (d\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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