Integrand size = 10, antiderivative size = 111 \[ \int \arccos (a+b x)^{5/2} \, dx=-\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{4 b} \]
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Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4888, 4716, 4768, 4810, 3385, 3433} \[ \int \arccos (a+b x)^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{4 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}-\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b} \]
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Rule 3385
Rule 3433
Rule 4716
Rule 4768
Rule 4810
Rule 4888
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \arccos (x)^{5/2} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {5 \text {Subst}\left (\int \frac {x \arccos (x)^{3/2}}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b} \\ & = -\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}-\frac {15 \text {Subst}\left (\int \sqrt {\arccos (x)} \, dx,x,a+b x\right )}{4 b} \\ & = -\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}-\frac {15 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx,x,a+b x\right )}{8 b} \\ & = -\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {15 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a+b x)\right )}{8 b} \\ & = -\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {15 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a+b x)}\right )}{4 b} \\ & = -\frac {15 (a+b x) \sqrt {\arccos (a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \arccos (a+b x)^{3/2}}{2 b}+\frac {(a+b x) \arccos (a+b x)^{5/2}}{b}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a+b x)}\right )}{4 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \arccos (a+b x)^{5/2} \, dx=-\frac {i \left (\sqrt {-i \arccos (a+b x)} \Gamma \left (\frac {7}{2},-i \arccos (a+b x)\right )-\sqrt {i \arccos (a+b x)} \Gamma \left (\frac {7}{2},i \arccos (a+b x)\right )\right )}{2 b \sqrt {\arccos (a+b x)}} \]
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Time = 0.95 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, b x -4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a +10 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, b x +15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, a -15 \pi \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )\right )}{8 b \sqrt {\pi }}\) | \(140\) |
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Exception generated. \[ \int \arccos (a+b x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \arccos (a+b x)^{5/2} \, dx=\int \operatorname {acos}^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
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Exception generated. \[ \int \arccos (a+b x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.65 \[ \int \arccos (a+b x)^{5/2} \, dx=\frac {\arccos \left (b x + a\right )^{\frac {5}{2}} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\arccos \left (b x + a\right )^{\frac {5}{2}} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {5 i \, \arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{4 \, b} - \frac {5 i \, \arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{4 \, b} - \frac {\left (15 i + 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{32 \, b} + \frac {\left (15 i - 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{32 \, b} - \frac {15 \, \sqrt {\arccos \left (b x + a\right )} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{8 \, b} - \frac {15 \, \sqrt {\arccos \left (b x + a\right )} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{8 \, b} \]
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Timed out. \[ \int \arccos (a+b x)^{5/2} \, dx=\int {\mathrm {acos}\left (a+b\,x\right )}^{5/2} \,d x \]
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