3.215 \(\int (d x)^{3/2} (a+b \cos ^{-1}(c x))^3 \, dx\)

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

[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 = 0.16, 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 = {} \[ \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

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 = 34.99, size = 0, normalized size = 0.00 \[ \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^3 \, dx \]

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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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} d x \arccos \left (c x\right )^{3} + 3 \, a b^{2} d x \arccos \left (c x\right )^{2} + 3 \, a^{2} b d x \arccos \left (c x\right ) + a^{3} d x\right )} \sqrt {d x}, x\right ) \]

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

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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maple [A]  time = 0.34, 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 \[ \frac {2}{5} \, b^{3} d^{\frac {3}{2}} x^{\frac {5}{2}} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )^{3} + \frac {1}{10} \, a^{3} c^{2} d^{\frac {3}{2}} {\left (\frac {4 \, {\left (c^{2} x^{\frac {5}{2}} + 5 \, \sqrt {x}\right )}}{c^{4}} - \frac {10 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {9}{2}}} + \frac {5 \, \log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {9}{2}}}\right )} + 15 \, a b^{2} c^{2} d^{\frac {3}{2}} \int \frac {x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )^{2}}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} + 15 \, a^{2} b c^{2} d^{\frac {3}{2}} \int \frac {x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 6 \, b^{3} c d^{\frac {3}{2}} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )^{2}}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - \frac {1}{2} \, a^{3} d^{\frac {3}{2}} {\left (\frac {4 \, \sqrt {x}}{c^{2}} - \frac {2 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} - 15 \, a b^{2} d^{\frac {3}{2}} \int \frac {x^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )^{2}}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 15 \, a^{2} b d^{\frac {3}{2}} \int \frac {x^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]

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

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

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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