Optimal. Leaf size=216 \[ \frac {15 b^2 \sqrt {a+b \text {ArcCos}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \text {ArcCos}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^{3/2}}{8 c}-\frac {(a+b \text {ArcCos}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^{5/2} \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 c^2} \]
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Rubi [A]
time = 0.43, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4726, 4796,
4738, 4810, 3393, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {15 \sqrt {\pi } b^{5/2} \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{128 c^2}+\frac {15 \sqrt {\pi } b^{5/2} \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^2 \sqrt {a+b \text {ArcCos}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \text {ArcCos}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^{3/2}}{8 c}-\frac {(a+b \text {ArcCos}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4726
Rule 4738
Rule 4796
Rule 4810
Rubi steps
\begin {align*} \int x \left (a+b \cos ^{-1}(c x)\right )^{5/2} \, dx &=\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {1}{4} (5 b c) \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac {1}{16} \left (15 b^2\right ) \int x \sqrt {a+b \cos ^{-1}(c x)} \, dx+\frac {(5 b) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac {1}{64} \left (15 b^3 c\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}} \, dx\\ &=-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{64 c^2}\\ &=-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{64 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \cos ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{128 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \cos ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{128 c^2}+\frac {\left (15 b^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{128 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \cos ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{64 c^2}+\frac {\left (15 b^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{64 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \cos ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^{5/2} \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 c^2}\\ \end {align*}
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Mathematica [A]
time = 1.39, size = 201, normalized size = 0.93 \begin {gather*} \frac {15 b^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi }}\right )+15 b^2 \sqrt {\pi } S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )+2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)} \left (\left (16 a^2-15 b^2\right ) \cos (2 \text {ArcCos}(c x))+16 b^2 \text {ArcCos}(c x)^2 \cos (2 \text {ArcCos}(c x))-20 a b \sin (2 \text {ArcCos}(c x))+4 b \text {ArcCos}(c x) (8 a \cos (2 \text {ArcCos}(c x))-5 b \sin (2 \text {ArcCos}(c x)))\right )}{128 \sqrt {\frac {1}{b}} c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs.
\(2(170)=340\).
time = 0.33, size = 414, normalized size = 1.92
method | result | size |
default | \(\frac {15 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}-15 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}+64 \arccos \left (c x \right )^{3} \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+192 \arccos \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+80 \arccos \left (c x \right )^{2} \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+192 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -60 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+160 \arccos \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+64 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-60 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+80 \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b}{256 c^{2} \sqrt {a +b \arccos \left (c x \right )}}\) | \(414\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, 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 = 1.20, size = 1307, normalized size = 6.05 \begin {gather*} \text {Too large to display} \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 = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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