Optimal. Leaf size=358 \[ \frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \]
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Rubi [A] time = 1.32, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {4630, 4708, 4678, 4620, 4724, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3351
Rule 3352
Rule 4620
Rule 4630
Rule 4678
Rule 4708
Rule 4724
Rubi steps
\begin {align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \, dx &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {1}{6} (5 b c) \int \frac {x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac {1}{12} \left (5 b^2\right ) \int x^2 \sqrt {a+b \cos ^{-1}(c x)} \, dx+\frac {(5 b) \int \frac {x \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{9 c}\\ &=-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^2\right ) \int \sqrt {a+b \cos ^{-1}(c x)} \, dx}{6 c^2}-\frac {1}{72} \left (5 b^3 c\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}} \, dx\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{72 c^3}-\frac {\left (5 b^3\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}} \, dx}{12 c}\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {a+b x}}+\frac {\cos (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{72 c^3}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac {\left (5 b^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}+\frac {\left (5 b^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {\left (5 b^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{6 c^3}+\frac {\left (5 b^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac {\left (5 b^3 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}+\frac {\left (5 b^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{6 c^3}+\frac {\left (5 b^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac {\left (5 b^3 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{6 c^3}+\frac {\left (5 b^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{48 c^3}+\frac {\left (5 b^2 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{144 c^3}+\frac {\left (5 b^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{48 c^3}+\frac {\left (5 b^2 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{144 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \cos ^{-1}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \cos ^{-1}(c x)}-\frac {5 b \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 c^3}\\ \end {align*}
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Mathematica [C] time = 17.23, size = 1002, normalized size = 2.80 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.42, size = 2760, normalized size = 7.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 792, normalized size = 2.21 \[ \frac {5 \sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b^{3}+5 \sqrt {3}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b^{3}+405 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b^{3}+405 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b^{3}+216 \arccos \left (c x \right )^{3} \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) b^{3}+72 \arccos \left (c x \right )^{3} \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) b^{3}+648 \arccos \left (c x \right )^{2} \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a \,b^{2}-540 \arccos \left (c x \right )^{2} \sin \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) b^{3}+216 \arccos \left (c x \right )^{2} \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a \,b^{2}-60 \arccos \left (c x \right )^{2} \sin \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) b^{3}+648 \arccos \left (c x \right ) \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a^{2} b -810 \arccos \left (c x \right ) \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) b^{3}-1080 \arccos \left (c x \right ) \sin \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a \,b^{2}+216 \arccos \left (c x \right ) \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) b^{3}-120 \arccos \left (c x \right ) \sin \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a \,b^{2}+216 \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a^{3}-810 \cos \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a \,b^{2}-540 \sin \left (\frac {a +b \arccos \left (c x \right )}{b}-\frac {a}{b}\right ) a^{2} b +72 \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a^{3}-30 \cos \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a \,b^{2}-60 \sin \left (\frac {3 a +3 b \arccos \left (c x \right )}{b}-\frac {3 a}{b}\right ) a^{2} b}{864 c^{3} \sqrt {a +b \arccos \left (c x \right )}} \]
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 \[ \int {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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