Optimal. Leaf size=111 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{4 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b} \]
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Rubi [A] time = 0.14736, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4804, 4620, 4678, 4724, 3304, 3352} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{4 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b} \]
Antiderivative was successfully verified.
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Rule 4804
Rule 4620
Rule 4678
Rule 4724
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \cos ^{-1}(a+b x)^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \cos ^{-1}(x)^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}+\frac{5 \operatorname{Subst}\left (\int \frac{x \cos ^{-1}(x)^{3/2}}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}-\frac{15 \operatorname{Subst}\left (\int \sqrt{\cos ^{-1}(x)} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}-\frac{15 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{8 b}\\ &=-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}+\frac{15 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a+b x)}\right )}{4 b}\\ &=-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{4 b}\\ \end{align*}
Mathematica [C] time = 0.0488174, size = 90, normalized size = 0.81 \[ -\frac{\frac{\sqrt{\cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{7}{2},-i \cos ^{-1}(a+b x)\right )}{2 \sqrt{-i \cos ^{-1}(a+b x)}}+\frac{\sqrt{\cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{7}{2},i \cos ^{-1}(a+b x)\right )}{2 \sqrt{i \cos ^{-1}(a+b x)}}}{b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.092, size = 140, normalized size = 1.3 \begin{align*} -{\frac{\sqrt{2}}{8\,b\sqrt{\pi }} \left ( -4\, \left ( \arccos \left ( bx+a \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }xb-4\, \left ( \arccos \left ( bx+a \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }a+10\, \left ( \arccos \left ( bx+a \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}+15\,\sqrt{2}\sqrt{\arccos \left ( bx+a \right ) }\sqrt{\pi }xb+15\,\sqrt{2}\sqrt{\arccos \left ( bx+a \right ) }\sqrt{\pi }a-15\,\pi \,{\it FresnelC} \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57023, size = 282, normalized size = 2.54 \begin{align*} \frac{5 \, i \arccos \left (b x + a\right )^{\frac{3}{2}} e^{\left (i \arccos \left (b x + a\right )\right )}}{4 \, b} + \frac{\arccos \left (b x + a\right )^{\frac{5}{2}} e^{\left (i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac{5 \, i \arccos \left (b x + a\right )^{\frac{3}{2}} e^{\left (-i \arccos \left (b x + a\right )\right )}}{4 \, b} + \frac{\arccos \left (b x + a\right )^{\frac{5}{2}} e^{\left (-i \arccos \left (b x + a\right )\right )}}{2 \, b} - \frac{15 \, \sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{16 \, b{\left (i - 1\right )}} - \frac{15 \, \sqrt{\arccos \left (b x + a\right )} e^{\left (i \arccos \left (b x + a\right )\right )}}{8 \, b} - \frac{15 \, \sqrt{\arccos \left (b x + a\right )} e^{\left (-i \arccos \left (b x + a\right )\right )}}{8 \, b} + \frac{15 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{16 \, b{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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