3.3.59 \(\int \frac {(c e+d e x)^4}{\sqrt {a+b \text {ArcSin}(c+d x)}} \, dx\) [259]

Optimal. Leaf size=365 \[ \frac {e^4 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d}-\frac {e^4 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 \sqrt {b} d}-\frac {e^4 \sqrt {\frac {3 \pi }{2}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{10}} S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{8 \sqrt {b} d} \]

[Out]

1/80*e^4*cos(5*a/b)*FresnelC(10^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*10^(1/2)*Pi^(1/2)/d/b^(1/2)+
1/80*e^4*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(5*a/b)*10^(1/2)*Pi^(1/2)/d/b^(1/2)+
1/8*e^4*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/b^(1/2)+1/8*e
^4*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/d/b^(1/2)-1/16*e^4*c
os(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/d/b^(1/2)-1/16*e^4*Fre
snelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/d/b^(1/2)

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Rubi [A]
time = 0.62, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4889, 12, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} e^4 \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d}-\frac {\sqrt {\frac {3 \pi }{2}} e^4 \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {\sqrt {\frac {\pi }{10}} e^4 \cos \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} e^4 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d}-\frac {\sqrt {\frac {3 \pi }{2}} e^4 \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {\sqrt {\frac {\pi }{10}} e^4 \sin \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4/Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

(e^4*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d) - (e^4*Sqrt
[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^4*Sqrt[
Pi/10]*Cos[(5*a)/b]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^4*Sqrt[Pi/
2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*Sqrt[b]*d) - (e^4*Sqrt[(3*Pi)/2]*Fr
esnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(8*Sqrt[b]*d) + (e^4*Sqrt[Pi/10]*Fresne
lS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(5*a)/b])/(8*Sqrt[b]*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 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 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*} \int \frac {(c e+d e x)^4}{\sqrt {a+b \sin ^{-1}(c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^4 x^4}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {\cos (x) \sin ^4(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \left (\frac {\cos (x)}{8 \sqrt {a+b x}}-\frac {3 \cos (3 x)}{16 \sqrt {a+b x}}+\frac {\cos (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}+\frac {e^4 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}+\frac {\left (e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac {\left (3 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}+\frac {\left (e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b d}-\frac {\left (3 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 b d}+\frac {\left (e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 b d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b d}-\frac {\left (3 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 b d}+\frac {\left (e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 b d}\\ &=\frac {e^4 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d}-\frac {e^4 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 \sqrt {b} d}-\frac {e^4 \sqrt {\frac {3 \pi }{2}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{8 \sqrt {b} d}+\frac {e^4 \sqrt {\frac {\pi }{10}} S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{8 \sqrt {b} d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.16, size = 370, normalized size = 1.01 \begin {gather*} \frac {i e^4 e^{-\frac {5 i a}{b}} \left (-10 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+10 e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+5 \sqrt {3} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )-5 \sqrt {3} e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )-\sqrt {5} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {5 i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {5} e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {5 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{160 d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^4/Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

((I/160)*e^4*(-10*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*
x]))/b] + 10*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] + 5
*Sqrt[3]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b]
- 5*Sqrt[3]*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b]
- Sqrt[5]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[5]*E^(
((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((5*I)
*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

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Maple [A]
time = 0.60, size = 317, normalized size = 0.87

method	result	size
default	\(\frac {e^{4} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, \left (\sqrt {-\frac {3}{b}}\, \sqrt {-\frac {5}{b}}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b -\sqrt {-\frac {3}{b}}\, \sqrt {-\frac {5}{b}}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b -2 \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {5}{b}}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +2 \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {5}{b}}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +\cos \left (\frac {5 a}{b}\right ) \FresnelC \left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )-\sin \left (\frac {5 a}{b}\right ) \mathrm {S}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )\right )}{80 d}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/80/d*e^4*2^(1/2)*Pi^(1/2)*(-5/b)^(1/2)*((-3/b)^(1/2)*(-5/b)^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3
/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-(-3/b)^(1/2)*(-5/b)^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/
b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-2*(-1/b)^(1/2)*(-5/b)^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^
(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+2*(-1/b)^(1/2)*(-5/b)^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/
2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+cos(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2
)/b)-sin(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^4/sqrt(b*arcsin(d*x + c) + a), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{4} \left (\int \frac {c^{4}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {d^{4} x^{4}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {4 c d^{3} x^{3}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {4 c^{3} d x}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4/(a+b*asin(d*x+c))**(1/2),x)

[Out]

e**4*(Integral(c**4/sqrt(a + b*asin(c + d*x)), x) + Integral(d**4*x**4/sqrt(a + b*asin(c + d*x)), x) + Integra
l(4*c*d**3*x**3/sqrt(a + b*asin(c + d*x)), x) + Integral(6*c**2*d**2*x**2/sqrt(a + b*asin(c + d*x)), x) + Inte
gral(4*c**3*d*x/sqrt(a + b*asin(c + d*x)), x))

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Giac [C] Result contains complex when optimal does not.
time = 0.86, size = 507, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {\pi } e^{4} \operatorname {erf}\left (-\frac {\sqrt {10} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} - \frac {i \, \sqrt {10} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (\frac {5 i \, a}{b}\right )}}{16 \, {\left (\sqrt {10} \sqrt {b} + \frac {i \, \sqrt {10} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} d} + \frac {\sqrt {6} \sqrt {\pi } e^{4} \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} - \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (\frac {3 i \, a}{b}\right )}}{32 \, \sqrt {b} d {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {\sqrt {\pi } e^{4} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{8 \, d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } e^{4} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{8 \, d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {6} \sqrt {\pi } e^{4} \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} + \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (-\frac {3 i \, a}{b}\right )}}{32 \, \sqrt {b} d {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {\sqrt {\pi } e^{4} \operatorname {erf}\left (-\frac {\sqrt {10} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} + \frac {i \, \sqrt {10} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (-\frac {5 i \, a}{b}\right )}}{16 \, {\left (\sqrt {10} \sqrt {b} - \frac {i \, \sqrt {10} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/16*sqrt(pi)*e^4*erf(-1/2*sqrt(10)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(10)*sqrt(b*arcsin(d*x +
c) + a)*sqrt(b)/abs(b))*e^(5*I*a/b)/((sqrt(10)*sqrt(b) + I*sqrt(10)*b^(3/2)/abs(b))*d) + 1/32*sqrt(6)*sqrt(pi)
*e^4*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/
abs(b))*e^(3*I*a/b)/(sqrt(b)*d*(I*b/abs(b) + 1)) - 1/8*sqrt(pi)*e^4*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c)
+ a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(d*(I*sqrt(2)*b/sqrt(abs
(b)) + sqrt(2)*sqrt(abs(b)))) - 1/8*sqrt(pi)*e^4*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) -
1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(d*(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqr
t(abs(b)))) + 1/32*sqrt(6)*sqrt(pi)*e^4*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*s
qrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(b)*d*(-I*b/abs(b) + 1)) - 1/16*sqrt(pi)*e^4*erf(
-1/2*sqrt(10)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(10)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))
*e^(-5*I*a/b)/((sqrt(10)*sqrt(b) - I*sqrt(10)*b^(3/2)/abs(b))*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^4/(a + b*asin(c + d*x))^(1/2),x)

[Out]

int((c*e + d*e*x)^4/(a + b*asin(c + d*x))^(1/2), x)

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