3.88 \(\int x^2 \cos ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=178 \[ \frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)} \]

[Out]

1/3*x^3*arccos(a*x)^(5/2)+5/864*FresnelC(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+15/32*Fresne
lC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3-5/9*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a^3-5/18*
x^2*arccos(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a-5/6*x*arccos(a*x)^(1/2)/a^2-5/36*x^3*arccos(a*x)^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4630, 4708, 4678, 4620, 4724, 3304, 3352, 3312} \[ \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCos[a*x]^(5/2),x]

[Out]

(-5*x*Sqrt[ArcCos[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcCos[a*x]])/36 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(9*a
^3) - (5*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(18*a) + (x^3*ArcCos[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelC
[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^3) + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(144*a^3)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[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]

Rule 4630

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m
+ 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4678

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] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(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]

Rule 4708

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rubi steps

\begin {align*} \int x^2 \cos ^{-1}(a x)^{5/2} \, dx &=\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {1}{6} (5 a) \int \frac {x^3 \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5}{12} \int x^2 \sqrt {\cos ^{-1}(a x)} \, dx+\frac {5 \int \frac {x \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac {5 \int \sqrt {\cos ^{-1}(a x)} \, dx}{6 a^2}-\frac {1}{72} (5 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{72 a^3}-\frac {5 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx}{12 a}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{72 a^3}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{288 a^3}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{96 a^3}+\frac {5 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {5 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{6 a^3}+\frac {5 \operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}+\frac {5 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac {5 x \sqrt {\cos ^{-1}(a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^3}+\frac {5 \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{144 a^3}\\ \end {align*}

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Mathematica [C]  time = 0.13, size = 122, normalized size = 0.69 \[ -\frac {81 i \sqrt {\cos ^{-1}(a x)^2} \Gamma \left (\frac {7}{2},-i \cos ^{-1}(a x)\right )+81 \cos ^{-1}(a x) \Gamma \left (\frac {7}{2},i \cos ^{-1}(a x)\right )+\sqrt {3} \left (i \sqrt {\cos ^{-1}(a x)^2} \Gamma \left (\frac {7}{2},-3 i \cos ^{-1}(a x)\right )+\cos ^{-1}(a x) \Gamma \left (\frac {7}{2},3 i \cos ^{-1}(a x)\right )\right )}{648 a^3 \sqrt {i \cos ^{-1}(a x)} \sqrt {\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*ArcCos[a*x]^(5/2),x]

[Out]

-1/648*((81*I)*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 81*ArcCos[a*x]*Gamma[7/2, I*ArcCos[a*x]] + S
qrt[3]*(I*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-3*I)*ArcCos[a*x]] + ArcCos[a*x]*Gamma[7/2, (3*I)*ArcCos[a*x]]))/(a^
3*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]])

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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.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [B]  time = 2.97, size = 364, normalized size = 2.04 \[ \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{144 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac {5 \, i \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{144 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} - \frac {5 \, \sqrt {6} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{1728 \, a^{3} {\left (i - 1\right )}} - \frac {15 \, \sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{64 \, a^{3} {\left (i - 1\right )}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{288 \, a^{3}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac {5 \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{288 \, a^{3}} + \frac {5 \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{1728 \, a^{3} {\left (i - 1\right )}} + \frac {15 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{64 \, a^{3} {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

5/144*i*arccos(a*x)^(3/2)*e^(3*i*arccos(a*x))/a^3 + 1/24*arccos(a*x)^(5/2)*e^(3*i*arccos(a*x))/a^3 + 5/16*i*ar
ccos(a*x)^(3/2)*e^(i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(5/2)*e^(i*arccos(a*x))/a^3 - 5/16*i*arccos(a*x)^(3/2)
*e^(-i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(5/2)*e^(-i*arccos(a*x))/a^3 - 5/144*i*arccos(a*x)^(3/2)*e^(-3*i*arc
cos(a*x))/a^3 + 1/24*arccos(a*x)^(5/2)*e^(-3*i*arccos(a*x))/a^3 - 5/1728*sqrt(6)*sqrt(pi)*i*erf(sqrt(6)*sqrt(a
rccos(a*x))/(i - 1))/(a^3*(i - 1)) - 15/64*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i -
1)) - 5/288*sqrt(arccos(a*x))*e^(3*i*arccos(a*x))/a^3 - 15/32*sqrt(arccos(a*x))*e^(i*arccos(a*x))/a^3 - 15/32
*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a^3 - 5/288*sqrt(arccos(a*x))*e^(-3*i*arccos(a*x))/a^3 + 5/1728*sqrt(6)*
sqrt(pi)*erf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 15/64*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(
arccos(a*x))/(i - 1))/(a^3*(i - 1))

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maple [A]  time = 0.24, size = 156, normalized size = 0.88 \[ \frac {216 a x \arccos \left (a x \right )^{3}+5 \sqrt {3}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+72 \arccos \left (a x \right )^{3} \cos \left (3 \arccos \left (a x \right )\right )+405 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-540 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-60 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-810 a x \arccos \left (a x \right )-30 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )}{864 a^{3} \sqrt {\arccos \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccos(a*x)^(5/2),x)

[Out]

1/864/a^3*(216*a*x*arccos(a*x)^3+5*3^(1/2)*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2
)*arccos(a*x)^(1/2))+72*arccos(a*x)^3*cos(3*arccos(a*x))+405*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2^(1/
2)/Pi^(1/2)*arccos(a*x)^(1/2))-540*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)-60*arccos(a*x)^2*sin(3*arccos(a*x))-810*a*
x*arccos(a*x)-30*arccos(a*x)*cos(3*arccos(a*x)))/arccos(a*x)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*acos(a*x)^(5/2),x)

[Out]

int(x^2*acos(a*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acos(a*x)**(5/2),x)

[Out]

Integral(x**2*acos(a*x)**(5/2), x)

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