3.3.77 \(\int \frac {(c e+d e x)^2}{(a+b \text {ArcSin}(c+d x))^{7/2}} \, dx\) [277]

Optimal. Leaf size=441 \[ -\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {2 e^2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {6 e^2 \sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {2 e^2 \sqrt {2 \pi } \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d}+\frac {6 e^2 \sqrt {6 \pi } \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{5 b^{7/2} d} \]

[Out]

-8/15*e^2*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+4/5*e^2*(d*x+c)^3/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+2/15*e^2*c
os(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d-2/15*e^2*Fresn
elC(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)/b^(7/2)/d-6/5*e^2*cos(3*a/b)
*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(7/2)/d+6/5*e^2*FresnelC(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)/b^(7/2)/d-2/5*e^2*(d*x+c)^2*(1-(d*
x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^(5/2)-16/15*e^2*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)+24/5
*e^2*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)

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Rubi [A]
time = 0.78, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4889, 12, 4729, 4807, 4727, 3387, 3386, 3432, 3385, 3433, 4717, 4809} \begin {gather*} -\frac {2 \sqrt {2 \pi } e^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {6 \sqrt {6 \pi } e^2 \sin \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 )}{5 b^{7/2} d}+\frac {2 \sqrt {2 \pi } e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {6 \sqrt {6 \pi } e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {24 e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {2 e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(7/2),x]

[Out]

(-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) - (8*e^2*(c + d*x))/(15*b^2*d
*(a + b*ArcSin[c + d*x])^(3/2)) + (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) - (16*e^2*Sqrt[1
- (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (24*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(5*b^3*d*
Sqrt[a + b*ArcSin[c + d*x]]) + (2*e^2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sq
rt[b]])/(15*b^(7/2)*d) - (6*e^2*Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt
[b]])/(5*b^(7/2)*d) - (2*e^2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(
15*b^(7/2)*d) + (6*e^2*Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(5*
b^(7/2)*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 4717

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

Rule 4727

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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 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 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[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]

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)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}-\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {\left (8 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}-\frac {\left (12 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d}-\frac {\left (24 e^2\right ) \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {a+b x}}+\frac {3 \sin (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (18 e^2\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (16 e^2 \cos \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 )}{15 b^3 d}+\frac {\left (6 e^2 \cos \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 )}{5 b^3 d}-\frac {\left (18 e^2 \cos \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 )}{5 b^3 d}+\frac {\left (16 e^2 \sin \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 )}{15 b^3 d}-\frac {\left (6 e^2 \sin \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 )}{5 b^3 d}+\frac {\left (18 e^2 \sin \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 )}{5 b^3 d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (32 e^2 \cos \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 )}{15 b^4 d}+\frac {\left (12 e^2 \cos \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 )}{5 b^4 d}-\frac {\left (36 e^2 \cos \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 )}{5 b^4 d}+\frac {\left (32 e^2 \sin \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 )}{15 b^4 d}-\frac {\left (12 e^2 \sin \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 )}{5 b^4 d}+\frac {\left (36 e^2 \sin \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 )}{5 b^4 d}\\ &=-\frac {2 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^2 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {24 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {2 e^2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {6 e^2 \sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {2 e^2 \sqrt {2 \pi } C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d}+\frac {6 e^2 \sqrt {6 \pi } C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{5 b^{7/2} d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.20, size = 538, normalized size = 1.22 \begin {gather*} \frac {e^2 \left (-3 b^2 e^{i \text {ArcSin}(c+d x)}+3 b^2 e^{3 i \text {ArcSin}(c+d x)}+2 e^{-\frac {i a}{b}} (a+b \text {ArcSin}(c+d x)) \left (e^{\frac {i (a+b \text {ArcSin}(c+d x))}{b}} (2 a-i b+2 b \text {ArcSin}(c+d x))-2 i b \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )+e^{-i \text {ArcSin}(c+d x)} \left (4 a^2+2 a b (i+4 \text {ArcSin}(c+d x))+b^2 \left (-3+2 i \text {ArcSin}(c+d x)+4 \text {ArcSin}(c+d x)^2\right )-4 e^{\frac {i (a+b \text {ArcSin}(c+d x))}{b}} (a+b \text {ArcSin}(c+d x))^2 \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 )\right )-6 e^{-\frac {3 i a}{b}} (a+b \text {ArcSin}(c+d x)) \left (e^{\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}} (6 a-i b+6 b \text {ArcSin}(c+d x))-6 i \sqrt {3} b \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )+3 e^{-3 i \text {ArcSin}(c+d x)} \left (b^2-2 (a+b \text {ArcSin}(c+d x)) \left (6 a+i b+6 b \text {ArcSin}(c+d x)+6 i \sqrt {3} b e^{\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )\right )}{60 b^3 d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(7/2),x]

[Out]

(e^2*(-3*b^2*E^(I*ArcSin[c + d*x]) + 3*b^2*E^((3*I)*ArcSin[c + d*x]) + (2*(a + b*ArcSin[c + d*x])*(E^((I*(a +
b*ArcSin[c + d*x]))/b)*(2*a - I*b + 2*b*ArcSin[c + d*x]) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Ga
mma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]))/E^((I*a)/b) + (4*a^2 + 2*a*b*(I + 4*ArcSin[c + d*x]) + b^2*(-3 +
(2*I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 4*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*S
qrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]) - (6*(a +
b*ArcSin[c + d*x])*(E^(((3*I)*(a + b*ArcSin[c + d*x]))/b)*(6*a - I*b + 6*b*ArcSin[c + d*x]) - (6*I)*Sqrt[3]*b*
(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b]))/E^(((3*I)*a)/b) + (3
*(b^2 - 2*(a + b*ArcSin[c + d*x])*(6*a + I*b + 6*b*ArcSin[c + d*x] + (6*I)*Sqrt[3]*b*E^(((3*I)*(a + b*ArcSin[c
+ d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/E^((3*I)*A
rcSin[c + d*x])))/(60*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1246\) vs. \(2(367)=734\).
time = 0.57, size = 1247, normalized size = 2.83

method	result	size
default	\(\text {Expression too large to display}\)	\(1247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/30/d*e^2/b^3*(36*arcsin(d*x+c)^2*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/
2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*b^2+36*arcsin(d*x+c)^2*(-3/b)^(1/2)*(a+b*arcsin(
d*x+c))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2
)*b^2-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(
1/2)/b)*cos(a/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b^2-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*Fresne
lC(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)*b^2+72*arcsin(d*x+
c)*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+
c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*a*b+72*arcsin(d*x+c)*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(3*a/b)*FresnelC(
3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*a*b-8*arcsin(d*x+c)*(a+b*arcsin(
d*x+c))^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*cos(a/b)*2^(1/2)*Pi^(1/2)*(-
1/b)^(1/2)*a*b-8*arcsin(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*FresnelC(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+36*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(3*a/b)
*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*a^2+36*(-3/b)^(1/2)*(a
+b*arcsin(d*x+c))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/
2)*Pi^(1/2)*a^2-4*(a+b*arcsin(d*x+c))^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b
)*cos(a/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a^2-4*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*FresnelC(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^2+4*arcsin(d*x+c)^2*cos(-(a+b*arcsin(d
*x+c))/b+a/b)*b^2-36*arcsin(d*x+c)^2*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^2+8*arcsin(d*x+c)*cos(-(a+b*arcsin(
d*x+c))/b+a/b)*a*b-2*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^2-72*arcsin(d*x+c)*cos(-3*(a+b*arcsin(d*x
+c))/b+3*a/b)*a*b+6*arcsin(d*x+c)*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^2+4*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^
2-3*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2-2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b-36*cos(-3*(a+b*arcsin(d*x+c))/b+
3*a/b)*a^2+3*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^2+6*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b)/(a+b*arcsin(d*
x+c))^(5/2)

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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)^2/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^2/(b*arcsin(d*x + c) + a)^(7/2), 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)^2/(a+b*arcsin(d*x+c))^(7/2),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^{2} \left (\int \frac {c^{2}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\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)**2/(a+b*asin(d*x+c))**(7/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2/(b*arcsin(d*x + c) + a)^(7/2), x)

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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 )}^2}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^2/(a + b*asin(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^2/(a + b*asin(c + d*x))^(7/2), x)

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