3.3.17 \(\int \frac {(a+b \text {ArcCos}(c x))^3}{\sqrt {d x}} \, dx\) [217]

Optimal. Leaf size=65 \[ \frac {2 \sqrt {d x} (a+b \text {ArcCos}(c x))^3}{d}+\frac {6 b c \text {Int}\left (\frac {\sqrt {d x} (a+b \text {ArcCos}(c x))^2}{\sqrt {1-c^2 x^2}},x\right )}{d} \]

[Out]

2*(a+b*arccos(c*x))^3*(d*x)^(1/2)/d+6*b*c*Unintegrable((a+b*arccos(c*x))^2*(d*x)^(1/2)/(-c^2*x^2+1)^(1/2),x)/d

________________________________________________________________________________________

Rubi [A]
time = 0.10, 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}{\sqrt {d x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 117.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcCos}(c x))^3}{\sqrt {d x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{\sqrt {d x}}\, dx\]

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)

________________________________________________________________________________________

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]

1/2*(4*b^3*sqrt(x)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + (a^3*c^2*sqrt(d)*(4*sqrt(x)/(c^2*d) - 2*arct
an(sqrt(c)*sqrt(x))/(c^(5/2)*d) + log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(c^(5/2)*d)) + 6*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*x^3 - d*x), x) + 6*a^2*b*c^2*sqr
t(d)*integrate(x^(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d*x^3 - d*x), x) - 12*b^3*c*sqrt(d)*int
egrate(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*x^3 - d*x), x)
+ a^3*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/(sqrt(c)*d) - log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(sqrt
(c)*d)) - 6*a*b^2*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*d*x^3 - d*x), x)
- 6*a^2*b*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d*x^3 - d*x), x))*sqrt(d)
)/sqrt(d)

________________________________________________________________________________________

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)/(d*x), x)

________________________________________________________________________________________

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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\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)

________________________________________________________________________________________