Integrand size = 16, antiderivative size = 358 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=-\frac {5 b^2 x \sqrt {a+b \arccos (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 c^3} \]
1/3*x^3*(a+b*arccos(c*x))^(5/2)+5/864*b^(5/2)*cos(3*a/b)*FresnelC(6^(1/2)/ Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/c^3+5/864*b^(5/ 2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*6 ^(1/2)*Pi^(1/2)/c^3+15/32*b^(5/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b* arccos(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^3+15/32*b^(5/2)*FresnelS(2^ (1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/ c^3-5/9*b*(a+b*arccos(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c^3-5/18*b*x^2*(a+b*a rccos(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c-5/6*b^2*x*(a+b*arccos(c*x))^(1/2)/c ^2-5/36*b^2*x^3*(a+b*arccos(c*x))^(1/2)
Result contains complex when optimal does not.
Time = 10.53 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.67 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=-\frac {i a^2 b e^{-\frac {3 i a}{b}} \left (-9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )+\sqrt {3} \left (-\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arccos (c x)}}-\frac {a \sqrt {b} \left (18 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (3 \sqrt {1-c^2 x^2}-2 c x \arccos (c x)\right )-9 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )-9 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )-\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )-\sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {3 a}{b}\right )-b \sin \left (\frac {3 a}{b}\right )\right )+6 \sqrt {b} \sqrt {a+b \arccos (c x)} (-2 \arccos (c x) \cos (3 \arccos (c x))+\sin (3 \arccos (c x)))\right )}{72 c^3}-\frac {\sqrt {b} \left (27 \left (2 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (-2 \sqrt {1-c^2 x^2} (a-5 b \arccos (c x))-b c x \left (-15+4 \arccos (c x)^2\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (\left (4 a^2-15 b^2\right ) \cos \left (\frac {a}{b}\right )-12 a b \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (12 a b \cos \left (\frac {a}{b}\right )+\left (4 a^2-15 b^2\right ) \sin \left (\frac {a}{b}\right )\right )\right )+\sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (\left (12 a^2-5 b^2\right ) \cos \left (\frac {3 a}{b}\right )-12 a b \sin \left (\frac {3 a}{b}\right )\right )+\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (12 a b \cos \left (\frac {3 a}{b}\right )+\left (12 a^2-5 b^2\right ) \sin \left (\frac {3 a}{b}\right )\right )+6 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (-b \left (-5+12 \arccos (c x)^2\right ) \cos (3 \arccos (c x))-2 (a-5 b \arccos (c x)) \sin (3 \arccos (c x))\right )\right )}{864 c^3} \]
((-1/72*I)*a^2*b*(-9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Ga mma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b* ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(-(Sqrt[( (-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b]) + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b* ArcCos[c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]]) - (a*Sqr t[b]*(18*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcC os[c*x]) - 9*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt [b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqr t[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) - Sqrt[6*Pi]* FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(b*Cos[(3*a)/b] + 2 *a*Sin[(3*a)/b]) - Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]] )/Sqrt[b]]*(2*a*Cos[(3*a)/b] - b*Sin[(3*a)/b]) + 6*Sqrt[b]*Sqrt[a + b*ArcC os[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(72*c^ 3) - (Sqrt[b]*(27*(2*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(-2*Sqrt[1 - c^2*x^2] *(a - 5*b*ArcCos[c*x]) - b*c*x*(-15 + 4*ArcCos[c*x]^2)) + Sqrt[2*Pi]*Fresn elC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*((4*a^2 - 15*b^2)*Cos[a/ b] - 12*a*b*Sin[a/b]) + Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[ c*x]])/Sqrt[b]]*(12*a*b*Cos[a/b] + (4*a^2 - 15*b^2)*Sin[a/b])) + Sqrt[6*Pi ]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*((12*a^2 - 5*b...
Time = 2.61 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.32, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5141, 5211, 5141, 5183, 5131, 5225, 3042, 3787, 25, 3042, 3785, 3786, 3793, 2009, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 (a+b \arccos (c x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {5}{6} b c \int \frac {x^3 (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int x^2 \sqrt {a+b \arccos (c x)}dx}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \int \sqrt {a+b \arccos (c x)}dx}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5131 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (\frac {1}{2} b c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+x \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos ^3\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \left (\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}+\frac {3 \cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\) |
(x^3*(a + b*ArcCos[c*x])^(5/2))/3 + (5*b*c*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/c^2 + (2*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x] )^(3/2))/c^2) - (3*b*(x*Sqrt[a + b*ArcCos[c*x]] - (Sqrt[b]*Sqrt[2*Pi]*Cos[ a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] + Sqrt[b]*Sqrt [2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2 *c)))/(2*c)))/(3*c^2) - (b*((x^3*Sqrt[a + b*ArcCos[c*x]])/3 - ((3*Sqrt[b]* Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] )/2 + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*Arc Cos[c*x]])/Sqrt[b]])/2 + (3*Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/2 + (Sqrt[b]*Sqrt[Pi/6]*FresnelS[(Sq rt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(6*c^3)))/(2*c )))/6
3.2.83.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcCos[c*x])^n/(m + 1)), x] + Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(797\) vs. \(2(278)=556\).
Time = 2.32 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.23
1/864/c^3/(a+b*arccos(c*x))^(1/2)*(405*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c *x))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x ))^(1/2)/b)*2^(1/2)*b^3-405*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)* sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b) *2^(1/2)*b^3+5*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(3*a/b)*FresnelC(3*2^(1 /2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)* b^3-5*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1 /2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*b^3+216*a rccos(c*x)^3*cos(-(a+b*arccos(c*x))/b+a/b)*b^3+72*arccos(c*x)^3*cos(-3*(a+ b*arccos(c*x))/b+3*a/b)*b^3+648*arccos(c*x)^2*cos(-(a+b*arccos(c*x))/b+a/b )*a*b^2+540*arccos(c*x)^2*sin(-(a+b*arccos(c*x))/b+a/b)*b^3+216*arccos(c*x )^2*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*a*b^2+60*arccos(c*x)^2*sin(-3*(a+b*a rccos(c*x))/b+3*a/b)*b^3+648*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*a^2 *b-810*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*b^3+1080*arccos(c*x)*sin( -(a+b*arccos(c*x))/b+a/b)*a*b^2+216*arccos(c*x)*cos(-3*(a+b*arccos(c*x))/b +3*a/b)*a^2*b-30*arccos(c*x)*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*b^3+120*arc cos(c*x)*sin(-3*(a+b*arccos(c*x))/b+3*a/b)*a*b^2+216*cos(-(a+b*arccos(c*x) )/b+a/b)*a^3-810*cos(-(a+b*arccos(c*x))/b+a/b)*a*b^2+540*sin(-(a+b*arccos( c*x))/b+a/b)*a^2*b+72*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*a^3-30*cos(-3*(a+b *arccos(c*x))/b+3*a/b)*a*b^2+60*sin(-3*(a+b*arccos(c*x))/b+3*a/b)*a^2*b...
Exception generated. \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\int x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \]
Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 2778, normalized size of antiderivative = 7.76 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \]
-1/576*(72*I*sqrt(2)*sqrt(pi)*a^3*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x ) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)* e^(I*a/b)/(I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 72*I*sqrt(2)*sqrt(pi)* a^3*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt( 2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 216*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqr t(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sq rt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 216*sqrt(2 )*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/s qrt(abs(b)) + b*sqrt(abs(b))) - 24*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x) ^2*e^(3*I*arccos(c*x)) - 72*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(I *arccos(c*x)) - 72*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(-I*arccos( c*x)) - 24*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(-3*I*arccos(c*x)) - 144*I*sqrt(pi)*a^3*b*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6) *b^(3/2) + I*sqrt(6)*b^(5/2)/abs(b)) + 144*I*sqrt(pi)*a^3*b*erf(-1/2*sqrt( 6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a) *sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b)) + 144*I*sqrt(pi)*a^3*sqrt(b)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/...
Timed out. \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]