\(\int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx\) [199]

Optimal result

Integrand size = 14, antiderivative size = 180 \[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} c^2} \]

[Out]

-8/3*cos(2*a/b)*FresnelS(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2)/c^2+8/3*FresnelC(2*(a+b*
arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(5/2)/c^2-2/3*x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(
c*x))^(3/2)-4/3/b^2/c^2/(a+b*arcsin(c*x))^(1/2)+8/3*x^2/b^2/(a+b*arcsin(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4729, 4807, 4731, 4491, 12, 3387, 3386, 3432, 3385, 3433, 4737} \[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {8 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}} \]

[In]

Int[x/(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(-2*x*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2)) - 4/(3*b^2*c^2*Sqrt[a + b*ArcSin[c*x]]) + (8*x^2)/(
3*b^2*Sqrt[a + b*ArcSin[c*x]]) - (8*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[P
i])])/(3*b^(5/2)*c^2) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*
b^(5/2)*c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3385

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 3386

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

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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]

Rule 3432

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]

Rule 3433

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4729

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

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4807

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}} \, dx}{3 b c}-\frac {(4 c) \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}} \, dx}{3 b} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {16 \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx}{3 b^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}+\frac {16 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}+\frac {16 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\left (8 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^2}+\frac {\left (8 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\left (16 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 b^3 c^2}+\frac {\left (16 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 b^3 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arcsin (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} c^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96 \[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c x)) \left (e^{-2 i \arcsin (c x)}+e^{2 i \arcsin (c x)}-\sqrt {2} e^{-\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )-\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )+b \sin (2 \arcsin (c x))}{3 b^2 c^2 (a+b \arcsin (c x))^{3/2}} \]

[In]

Integrate[x/(a + b*ArcSin[c*x])^(5/2),x]

[Out]

-1/3*(2*(a + b*ArcSin[c*x])*(E^((-2*I)*ArcSin[c*x]) + E^((2*I)*ArcSin[c*x]) - (Sqrt[2]*Sqrt[((-I)*(a + b*ArcSi
n[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a +
b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b]) + b*Sin[2*ArcSin[c*x]])/(b^2*c^2*(a + b*ArcSin[
c*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(142)=284\).

Time = 0.07 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.89

method	result	size
default	\(-\frac {-8 \arcsin \left (c x \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b -8 \arcsin \left (c x \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b -8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, a -8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, a +4 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b -\sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b +4 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a}{3 c^{2} b^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\)	\(340\)

[In]

int(x/(a+b*arcsin(c*x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/c^2/b^2*(-8*arcsin(c*x)*(-1/b)^(1/2)*Pi^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*ar
csin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*b-8*arcsin(c*x)*(-1/b)^(1/2)*Pi^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2
)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*b-8*(-1/b)^(1/2)*Pi^(1/2)*cos(2*a/b
)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*a-8*(-1/b)^(1/2)
*Pi^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/
2)*a+4*arcsin(c*x)*cos(-2*(a+b*arcsin(c*x))/b+2*a/b)*b-sin(-2*(a+b*arcsin(c*x))/b+2*a/b)*b+4*cos(-2*(a+b*arcsi
n(c*x))/b+2*a/b)*a)/(a+b*arcsin(c*x))^(3/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/(a+b*arcsin(c*x))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x/(a+b*asin(c*x))**(5/2),x)

[Out]

Integral(x/(a + b*asin(c*x))**(5/2), x)

Maxima [F]

\[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/(a+b*arcsin(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate(x/(b*arcsin(c*x) + a)^(5/2), x)

Giac [F]

\[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/(a+b*arcsin(c*x))^(5/2),x, algorithm="giac")

[Out]

integrate(x/(b*arcsin(c*x) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(x/(a + b*asin(c*x))^(5/2),x)

[Out]

int(x/(a + b*asin(c*x))^(5/2), x)