3.265 \(\int \frac{(c e+d e x)^4}{(a+b \sin ^{-1}(c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=412 \[ \frac{\sqrt{\frac{\pi }{2}} e^4 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 b^{3/2} d}-\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{\frac{5 \pi }{2}} e^4 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^4 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 b^{3/2} d}+\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{\sqrt{\frac{5 \pi }{2}} e^4 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

[Out]

(-2*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^4*Sqrt[Pi/2]*Cos[a/b]*Fresne
lS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d) + (3*e^4*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*Fresn
elS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) - (e^4*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*Fresne
lS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) + (e^4*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)*d) - (3*e^4*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*d) + (e^4*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*d)

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Rubi [A]  time = 0.866043, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4805, 12, 4631, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} e^4 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 b^{3/2} d}-\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{\frac{5 \pi }{2}} e^4 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^4 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 b^{3/2} d}+\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{\sqrt{\frac{5 \pi }{2}} e^4 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(-2*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^4*Sqrt[Pi/2]*Cos[a/b]*Fresne
lS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d) + (3*e^4*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*Fresn
elS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) - (e^4*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*Fresne
lS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) + (e^4*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)*d) - (3*e^4*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*d) + (e^4*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*d)

Rule 4805

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]

Rule 12

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

Rule 4631

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

Rule 3306

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 3305

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 3351

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

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 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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{8 \sqrt{a+b x}}+\frac{9 \sin (3 x)}{16 \sqrt{a+b x}}-\frac{5 \sin (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{e^4 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}-\frac{\left (5 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}+\frac{\left (9 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}\\ &=-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (e^4 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}+\frac{\left (9 e^4 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{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 )}{8 b d}-\frac{\left (5 e^4 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{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 )}{8 b d}+\frac{\left (e^4 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}-\frac{\left (9 e^4 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{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 )}{8 b d}+\frac{\left (5 e^4 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{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 )}{8 b d}\\ &=-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (e^4 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b^2 d}+\frac{\left (9 e^4 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}-\frac{\left (5 e^4 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{5 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}+\frac{\left (e^4 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b^2 d}-\frac{\left (9 e^4 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}+\frac{\left (5 e^4 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{5 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}\\ &=-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{e^4 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 b^{3/2} d}+\frac{3 e^4 \sqrt{\frac{3 \pi }{2}} \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 )}{4 b^{3/2} d}-\frac{e^4 \sqrt{\frac{5 \pi }{2}} \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{e^4 \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 b^{3/2} d}-\frac{3 e^4 \sqrt{\frac{3 \pi }{2}} 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 )}{4 b^{3/2} d}+\frac{e^4 \sqrt{\frac{5 \pi }{2}} C\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{5 a}{b}\right )}{4 b^{3/2} d}\\ \end{align*}

Mathematica [C]  time = 0.649081, size = 572, normalized size = 1.39 \[ \frac{e^4 e^{-\frac{5 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (2 e^{\frac{4 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 e^{\frac{6 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-3 \sqrt{3} e^{\frac{2 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-3 \sqrt{3} e^{\frac{8 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{5} e^{5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{5} e^{\frac{5 i \left (2 a+b \sin ^{-1}(c+d x)\right )}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{5 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+3 e^{\frac{5 i a}{b}+2 i \sin ^{-1}(c+d x)}-2 e^{\frac{5 i a}{b}+4 i \sin ^{-1}(c+d x)}-2 e^{\frac{5 i a}{b}+6 i \sin ^{-1}(c+d x)}+3 e^{\frac{5 i a}{b}+8 i \sin ^{-1}(c+d x)}-e^{\frac{5 i \left (a+2 b \sin ^{-1}(c+d x)\right )}{b}}-e^{\frac{5 i a}{b}}\right )}{16 b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(e^4*(-E^(((5*I)*a)/b) + 3*E^(((5*I)*a)/b + (2*I)*ArcSin[c + d*x]) - 2*E^(((5*I)*a)/b + (4*I)*ArcSin[c + d*x])
- 2*E^(((5*I)*a)/b + (6*I)*ArcSin[c + d*x]) + 3*E^(((5*I)*a)/b + (8*I)*ArcSin[c + d*x]) - E^(((5*I)*(a + 2*b*
ArcSin[c + d*x]))/b) + 2*E^(((4*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[
1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 2*E^(((6*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d
*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((
-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((8*I)*a)/b + (5
*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[
5]*E^((5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c + d*x])
)/b] + Sqrt[5]*E^(((5*I)*(2*a + b*ArcSin[c + d*x]))/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((5*I)*(
a + b*ArcSin[c + d*x]))/b]))/(16*b*d*E^(((5*I)*(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])

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Maple [A]  time = 0.117, size = 478, normalized size = 1.2 \begin{align*} -{\frac{{e}^{4}}{8\,bd} \left ( \sqrt{5}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ( 5\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sqrt{5}\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ( 5\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -3\,\sqrt{{b}^{-1}}\sqrt{3}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +3\,\sqrt{{b}^{-1}}\sqrt{3}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +2\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -2\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +2\,\cos \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) -3\,\cos \left ( 3\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-3\,{\frac{a}{b}} \right ) +\cos \left ( 5\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-5\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arcsin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^4/(b*arcsin(d*x + c) + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4/(a+b*asin(d*x+c))**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4/(b*arcsin(d*x + c) + a)^(3/2), x)