Optimal. Leaf size=65 \[ -\frac {6 b c \text {Int}\left (\frac {\left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2} \sqrt {d x}},x\right )}{d}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^3}{d \sqrt {d x}} \]
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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 = {} \[ \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{(d x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^3}{d \sqrt {d x}}-\frac {(6 b c) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d}\\ \end {align*}
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Mathematica [A] time = 8.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{(d x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \arccos \left (c x\right )^{3} + 3 \, a b^{2} \arccos \left (c x\right )^{2} + 3 \, a^{2} b \arccos \left (c x\right ) + a^{3}\right )} \sqrt {d x}}{d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{3}}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{\left (d x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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