Optimal. Leaf size=89 \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4804, 4620, 4678, 4624, 3305, 3351} \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4620
Rule 4624
Rule 4678
Rule 4804
Rubi steps
\begin {align*} \int \cos ^{-1}(a+b x)^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \cos ^{-1}(x)^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \operatorname {Subst}\left (\int \frac {x \sqrt {\cos ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{4 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 76, normalized size = 0.85 \[ -\frac {\sqrt {-i \cos ^{-1}(a+b x)} \Gamma \left (\frac {5}{2},-i \cos ^{-1}(a+b x)\right )+\sqrt {i \cos ^{-1}(a+b x)} \Gamma \left (\frac {5}{2},i \cos ^{-1}(a+b x)\right )}{2 b \sqrt {\cos ^{-1}(a+b x)}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [B] time = 0.39, size = 164, normalized size = 1.84 \[ \frac {3 \, i \sqrt {\arccos \left (b x + a\right )} e^{\left (i \arccos \left (b x + a\right )\right )}}{4 \, b} + \frac {\arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac {3 \, i \sqrt {\arccos \left (b x + a\right )} e^{\left (-i \arccos \left (b x + a\right )\right )}}{4 \, b} + \frac {\arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (-i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac {3 \, \sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (b x + a\right )}}{i - 1}\right )}{8 \, b {\left (i - 1\right )}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (b x + a\right )}}{i - 1}\right )}{8 \, b {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 105, normalized size = 1.18 \[ \frac {\sqrt {2}\, \left (2 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x b +2 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a -3 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+3 \pi \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )\right )}{4 b \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {acos}\left (a+b\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acos}^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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