Optimal. Leaf size=111 \[ -\frac {15 (a+b x) \sqrt {\text {ArcCos}(a+b x)}}{4 b}-\frac {5 \sqrt {1-(a+b x)^2} \text {ArcCos}(a+b x)^{3/2}}{2 b}+\frac {(a+b x) \text {ArcCos}(a+b x)^{5/2}}{b}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{4 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4888, 4716,
4768, 4810, 3385, 3433} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{4 b}+\frac {(a+b x) \text {ArcCos}(a+b x)^{5/2}}{b}-\frac {5 \sqrt {1-(a+b x)^2} \text {ArcCos}(a+b x)^{3/2}}{2 b}-\frac {15 (a+b x) \sqrt {\text {ArcCos}(a+b x)}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4716
Rule 4768
Rule 4810
Rule 4888
Rubi steps
\begin {align*} \int \cos ^{-1}(a+b x)^{5/2} \, dx &=\frac {\text {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 \text {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 \text {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 \text {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 \text {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 \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 90, normalized size = 0.81 \begin {gather*} -\frac {\frac {\sqrt {\text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {7}{2},-i \text {ArcCos}(a+b x)\right )}{2 \sqrt {-i \text {ArcCos}(a+b x)}}+\frac {\sqrt {\text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {7}{2},i \text {ArcCos}(a+b x)\right )}{2 \sqrt {i \text {ArcCos}(a+b x)}}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 140, normalized size = 1.26
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, b x -4 \arccos \left (b x +a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a +10 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, b x +15 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, a -15 \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )\right )}{8 b \sqrt {\pi }}\) | \(140\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \operatorname {acos}^{\frac {5}{2}}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.53, size = 183, normalized size = 1.65 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int {\mathrm {acos}\left (a+b\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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