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 = 0.09211, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, 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 4804
Rule 4620
Rule 4678
Rule 4624
Rule 3305
Rule 3351
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*}
Mathematica [C] time = 0.0353963, size = 76, normalized size = 0.85 \[ -\frac{\sqrt{-i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{5}{2},-i \cos ^{-1}(a+b x)\right )+\sqrt{i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{5}{2},i \cos ^{-1}(a+b x)\right )}{2 b \sqrt{\cos ^{-1}(a+b x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.082, size = 105, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2}}{4\,b\sqrt{\pi }} \left ( 2\, \left ( \arccos \left ( bx+a \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }xb+2\, \left ( \arccos \left ( bx+a \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }a-3\,\sqrt{2}\sqrt{\pi }\sqrt{\arccos \left ( bx+a \right ) }\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}+3\,\pi \,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( bx+a \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acos}^{\frac{3}{2}}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.62439, size = 221, normalized size = 2.48 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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