3.3.19 \(\int \frac {(a+b \text {ArcCos}(c x))^3}{(d x)^{5/2}} \, dx\) [219]

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

[Out]

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

Verification is not applicable to the result.

[In]

Int[(a + b*ArcCos[c*x])^3/(d*x)^(5/2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 68.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcCos}(c x))^3}{(d x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(5/2), x]

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Maple [A]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{\left (d x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))^3/(d*x)^(5/2),x)

[Out]

int((a+b*arccos(c*x))^3/(d*x)^(5/2),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)^(5/2),x, algorithm="maxima")

[Out]

-1/6*(4*b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + (3*a^3*c^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/(sqrt
(c)*d^3) - log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(sqrt(c)*d^3)) - 18*a*b^2*c^2*sqrt(d)*integrate(x^
(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*d^3*x^5 - d^3*x^3), x) - 18*a^2*b*c^2*sqrt(d)*integrat
e(x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^3*x^5 - d^3*x^3), x) - 12*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*d^3*x^5 - d^3*x^3), x)
- a^3*sqrt(d)*(6*c^(3/2)*arctan(sqrt(c)*sqrt(x))/d^3 - 3*c^(3/2)*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c
)))/d^3 - 4/(d^3*x^(3/2))) + 18*a*b^2*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(
c^2*d^3*x^5 - d^3*x^3), x) + 18*a^2*b*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^
2*d^3*x^5 - d^3*x^3), x))*d^(5/2)*x^(3/2))/(d^(5/2)*x^(3/2))

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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)^(5/2),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)/(d^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))**3/(d*x)**(5/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [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)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccos(c*x) + a)^3/(d*x)^(5/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))^3/(d*x)^(5/2),x)

[Out]

int((a + b*acos(c*x))^3/(d*x)^(5/2), x)

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