\(\int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx\) [271]

Optimal result

Integrand size = 25, antiderivative size = 344 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {4 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {4 e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{3 b^{5/2} d} \]

[Out]

-4/3*e^3*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2)/d+4/3*e^3*FresnelC
(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(5/2)/d+4/3*e^3*cos(4*a/b)*FresnelS(2*2^(
1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/d-4/3*e^3*FresnelC(2*2^(1/2)/Pi^(1/2
)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/b^(5/2)/d-2/3*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/
2)/b/d/(a+b*arcsin(d*x+c))^(3/2)-4*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^(1/2)+16/3*e^3*(d*x+c)^4/b^2/d/(a+b
*arcsin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4889, 12, 4729, 4807, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {4 \sqrt {\pi } e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}-\frac {4 \sqrt {2 \pi } e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {4 \sqrt {2 \pi } e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 \sqrt {\pi } e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b d (a+b \arcsin (c+d x))^{3/2}} \]

[In]

Int[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(5/2),x]

[Out]

(-2*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (4*e^3*(c + d*x)^2)/(b^2*d*
Sqrt[a + b*ArcSin[c + d*x]]) + (16*e^3*(c + d*x)^4)/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) + (4*e^3*Sqrt[2*Pi]*
Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) - (4*e^3*Sqrt[Pi]*Cos
[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d) + (4*e^3*Sqrt[Pi]*Fresne
lC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d) - (4*e^3*Sqrt[2*Pi]*Fresnel
C[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(3*b^(5/2)*d)

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 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]

Rule 4889

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^3 x^3}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{b d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{b^2 d}-\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (8 e^3\right ) \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+d x)\right )}{b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (8 e^3\right ) \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+d x)\right )}{b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}+\frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}+\frac {\left (16 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (4 e^3 \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+d x)\right )}{b^3 d}-\frac {\left (16 e^3 \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+d x)\right )}{3 b^3 d}+\frac {\left (8 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^3 \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+d x)\right )}{b^3 d}+\frac {\left (16 e^3 \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+d x)\right )}{3 b^3 d}-\frac {\left (8 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (8 e^3 \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+d x)}\right )}{b^3 d}-\frac {\left (32 e^3 \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+d x)}\right )}{3 b^3 d}+\frac {\left (16 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d}-\frac {\left (8 e^3 \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+d x)}\right )}{b^3 d}+\frac {\left (32 e^3 \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+d x)}\right )}{3 b^3 d}-\frac {\left (16 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {4 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {4 e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{3 b^{5/2} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.22 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.02 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {e^3 \left (-4 (a+b \arcsin (c+d x)) \left (e^{-2 i \arcsin (c+d x)}+e^{2 i \arcsin (c+d x)}-\sqrt {2} e^{-\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )+4 (a+b \arcsin (c+d x)) \left (e^{-4 i \arcsin (c+d x)}+e^{4 i \arcsin (c+d x)}-2 e^{-\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )-2 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )-2 b \sin (2 \arcsin (c+d x))+b \sin (4 \arcsin (c+d x))\right )}{12 b^2 d (a+b \arcsin (c+d x))^{3/2}} \]

[In]

Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(5/2),x]

[Out]

(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((-2*I)*ArcSin[c + d*x]) + E^((2*I)*ArcSin[c + d*x]) - (Sqrt[2]*Sqrt[((-I)
*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((2*
I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]) + 4*(a + b*ArcSin[
c + d*x])*(E^((-4*I)*ArcSin[c + d*x]) + E^((4*I)*ArcSin[c + d*x]) - (2*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*
Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((4*I)*a)/b) - 2*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c +
d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]) - 2*b*Sin[2*ArcSin[c + d*x]] + b*Sin[4*ArcSin[c + d*x
]]))/(12*b^2*d*(a + b*ArcSin[c + d*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(732\) vs. \(2(284)=568\).

Time = 1.30 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.13

method	result	size
default	\(\frac {e^{3} \left (-16 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \arcsin \left (d x +c \right ) b -16 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \arcsin \left (d x +c \right ) b +16 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +16 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -16 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, a -16 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, a +16 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a +16 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a -8 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +8 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) \arcsin \left (d x +c \right ) b +2 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -8 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a -\sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b +8 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \right )}{12 d \,b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}}\)	\(733\)

[In]

int((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/12*e^3/d/b^2*(-16*cos(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1
/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*arcsin(d*x+c)*b-16*sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b
)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*arcsin(d*x+c)*b+1
6*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^
(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+16*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/
b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-16*cos(4*a/b)*FresnelS(2*2^(1/2)/Pi
^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*a-16*
sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)
*(a+b*arcsin(d*x+c))^(1/2)*a+16*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/
Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a+16*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*
a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a-8*arcsin(d*x+c)*cos(-2*(a+b*arcsi
n(d*x+c))/b+2*a/b)*b+8*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*arcsin(d*x+c)*b+2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/
b)*b-8*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a-sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b+8*cos(-4*(a+b*arcsin(d*x+c)
)/b+4*a/b)*a)/(a+b*arcsin(d*x+c))^(3/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**3/(a+b*asin(d*x+c))**(5/2),x)

[Out]

e**3*(Integral(c**3/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqr
t(a + b*asin(c + d*x))*asin(c + d*x)**2), x) + Integral(d**3*x**3/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt
(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2), x) + Integral(3*c*d**2
*x**2/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c
+ d*x))*asin(c + d*x)**2), x) + Integral(3*c**2*d*x/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c
+ d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3/(b*arcsin(d*x + c) + a)^(5/2), x)

Giac [F]

\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3/(b*arcsin(d*x + c) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(5/2),x)

[Out]

int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(5/2), x)