3.1408 \(\int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx\)

Optimal. Leaf size=926 \[ -\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{4 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right ) d^{5/2}}{4 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right ) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt{2} b^3 f}-\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} d^2}{2 b f g}-\frac{a \sqrt{g \cos (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} d^2}{b^2 f \sqrt{\sin (2 e+2 f x)}} \]

[Out]

(a^2*d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[
2]*b^3*f) + (d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])]
)/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin
[e + f*x]])])/(Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sq
rt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqr
t[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*C
ot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + (a^2*d^(5/2)*Sqr
t[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqr
t[2]*b^3*f) + (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqr
t[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSi
n[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b
]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[
e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Si
n[e + f*x]]) - (d^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f*g) - (a*d^2*Sqrt[g*Cos[e + f*x]]*Ellip
ticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.84226, antiderivative size = 926, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 15, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.405, Rules used = {2909, 2568, 2575, 297, 1162, 617, 204, 1165, 628, 2572, 2639, 2906, 2905, 490, 1218} \[ -\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right ) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{4 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right ) d^{5/2}}{4 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right ) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{8 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \log \left (\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt{2} b^3 f}-\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} d^2}{2 b f g}-\frac{a \sqrt{g \cos (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} d^2}{b^2 f \sqrt{\sin (2 e+2 f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))/(a + b*Sin[e + f*x]),x]

[Out]

(a^2*d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[
2]*b^3*f) + (d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])]
)/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin
[e + f*x]])])/(Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sq
rt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqr
t[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*C
ot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + (a^2*d^(5/2)*Sqr
t[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqr
t[2]*b^3*f) + (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqr
t[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSi
n[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b
]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[
e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Si
n[e + f*x]]) - (d^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f*g) - (a*d^2*Sqrt[g*Cos[e + f*x]]*Ellip
ticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])

Rule 2909

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[(a*d)/b, Int[(
(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1))/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && N
eQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2575

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{k = Denomina
tor[m]}, -Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Si
n[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2906

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x
_)])), x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]
*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2905

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]))
, x_Symbol] :> Dist[(-4*Sqrt[2]*g)/f, Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sq
rt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx &=\frac{d \int \sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} \, dx}{b}-\frac{(a d) \int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx}{b}\\ &=-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{\left (a d^2\right ) \int \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} \, dx}{b^2}+\frac{\left (a^2 d^2\right ) \int \frac{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^2}+\frac{d^3 \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}} \, dx}{4 b}\\ &=-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}+\frac{\left (a^2 d^3\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}} \, dx}{b^3}-\frac{\left (a^3 d^3\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^3}-\frac{\left (d^4 g\right ) \operatorname{Subst}\left (\int \frac{x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 b f}-\frac{\left (a d^2 \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{b^2 \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{a d^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b^2 f \sqrt{\sin (2 e+2 f x)}}+\frac{\left (d^3 g\right ) \operatorname{Subst}\left (\int \frac{g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{4 b f}-\frac{\left (d^3 g\right ) \operatorname{Subst}\left (\int \frac{g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{4 b f}-\frac{\left (2 a^2 d^4 g\right ) \operatorname{Subst}\left (\int \frac{x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^3 f}-\frac{\left (a^3 d^3 \sqrt{\sin (e+f x)}\right ) \int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b^3 \sqrt{d \sin (e+f x)}}\\ &=-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{a d^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{\left (d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}+2 x}{-\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}-\frac{\left (d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}-2 x}{-\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}-\frac{\left (d^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 b f}-\frac{\left (d^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 b f}+\frac{\left (a^2 d^3 g\right ) \operatorname{Subst}\left (\int \frac{g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^3 f}-\frac{\left (a^2 d^3 g\right ) \operatorname{Subst}\left (\int \frac{g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{b^3 f}+\frac{\left (4 \sqrt{2} a^3 d^3 g \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^3 f \sqrt{d \sin (e+f x)}}\\ &=-\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}+\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{a d^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{\left (a^2 d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}+2 x}{-\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}-\frac{\left (a^2 d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{g}}{\sqrt{d}}-2 x}{-\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}-\frac{\left (d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}+\frac{\left (d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}-\frac{\left (a^2 d^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}-\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 b^3 f}-\frac{\left (a^2 d^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{g}{d}+\frac{\sqrt{2} \sqrt{g} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 b^3 f}+\frac{\left (2 \sqrt{2} a^3 d^3 g \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b} g-\sqrt{-a+b} x^2\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^3 \sqrt{-a+b} f \sqrt{d \sin (e+f x)}}-\frac{\left (2 \sqrt{2} a^3 d^3 g \sqrt{\sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b} g+\sqrt{-a+b} x^2\right ) \sqrt{1-\frac{x^4}{g^2}}} \, dx,x,\frac{\sqrt{g \cos (e+f x)}}{\sqrt{1+\sin (e+f x)}}\right )}{b^3 \sqrt{-a+b} f \sqrt{d \sin (e+f x)}}\\ &=\frac{d^{5/2} \sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}-\frac{d^{5/2} \sqrt{g} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}-\frac{a^2 d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}-\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}+\frac{a^2 d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}+\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}-\frac{2 \sqrt{2} a^3 d^3 \sqrt{g} \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^3 \sqrt{-a+b} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 d^3 \sqrt{g} \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^3 \sqrt{-a+b} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{a d^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{\left (a^2 d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^3 f}+\frac{\left (a^2 d^{5/2} \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^3 f}\\ &=\frac{a^2 d^{5/2} \sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^3 f}+\frac{d^{5/2} \sqrt{g} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}-\frac{a^2 d^{5/2} \sqrt{g} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{\sqrt{2} b^3 f}-\frac{d^{5/2} \sqrt{g} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right )}{4 \sqrt{2} b f}-\frac{a^2 d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}-\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}+\frac{a^2 d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{2 \sqrt{2} b^3 f}+\frac{d^{5/2} \sqrt{g} \log \left (\sqrt{g}+\sqrt{g} \cot (e+f x)+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right )}{8 \sqrt{2} b f}-\frac{2 \sqrt{2} a^3 d^3 \sqrt{g} \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^3 \sqrt{-a+b} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 d^3 \sqrt{g} \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt{\sin (e+f x)}}{b^3 \sqrt{-a+b} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}-\frac{d^2 (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{2 b f g}-\frac{a d^2 \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{b^2 f \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}

Mathematica [C]  time = 27.314, size = 1626, normalized size = 1.76 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))/(a + b*Sin[e + f*x]),x]

[Out]

-(Sqrt[g*Cos[e + f*x]]*Cot[e + f*x]*Csc[e + f*x]*(d*Sin[e + f*x])^(5/2))/(2*b*f) + (Sqrt[g*Cos[e + f*x]]*(d*Si
n[e + f*x])^(5/2)*((-2*b*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])
+ a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*S
qrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) -
(Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]
] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2
)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Ta
n[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]
^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a*Co
s[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[
Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -T
an[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1,
11/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*
ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]
*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*Ar
cTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1
+ (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt
[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt
[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqr
t[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 -
b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan
[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] +
Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt
[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 +
Tan[e + f*x]^2])))/(42*a*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*
Sqrt[1 + Tan[e + f*x]^2])))/(4*b*f*Sqrt[Cos[e + f*x]]*Sin[e + f*x]^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.391, size = 4649, normalized size = 5. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(f*x+e))^(5/2)*(g*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x)

[Out]

1/4/f*2^(1/2)*a/b^3/(-a^2+b^2)^(1/2)/(a-b+(-a^2+b^2)^(1/2))/(b+(-a^2+b^2)^(1/2)-a)*(a-b)*(4*I*cos(f*x+e)*(-(-1
+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f
*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)
*a^2+I*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)
*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^
(1/2))*(-a^2+b^2)^(1/2)*b^2-4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2
)),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+
e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3-4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a
/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e)
)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3+2*cos(f*x+e)^2*2^(1/2)*sin(f*x+e)*(-a^2+
b^2)^(1/2)*b^2-4*cos(f*x+e)^2*2^(1/2)*(-a^2+b^2)^(1/2)*a*b+4*cos(f*x+e)*2^(1/2)*(-a^2+b^2)^(1/2)*a*b-cos(f*x+e
)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
)/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2
)^(1/2)*b^2-4*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e)
)^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I
,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+s
in(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*
x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+2*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^
(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x
+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+I*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))
^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f
*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2-8*(-(-1+cos(f*x+e)-sin(f*x+e))
/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE(
(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b+4*(-(-1+cos(f*x+e)-sin(f*x+e)
)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF
((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b-4*I*(-(-1+cos(f*x+e)-sin(f*x
+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*Ellipt
icPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-I*(-(-1+cos(f*
x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^
(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+4*
I*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
)/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2
)^(1/2)*a^2+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*
cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+c
os(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2-4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),
a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2*b+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x
+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+
e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^
2+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*cos(f*x+e
)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
)/sin(f*x+e))^(1/2)*a^2*b-4*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+
e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(
1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-8*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*(
(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e)-sin
(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b+4*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x
+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+co
s(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b-4*I*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f
*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*Elli
pticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-I*cos(f*x+e)
*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))
/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)
^(1/2)*b^2+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*
cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+c
os(f*x+e))/sin(f*x+e))^(1/2)*a^3+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)
^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1
/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2-4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+
e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2*b+4*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+
e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((
-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2+4*Elliptic
Pi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(
f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2
*b-4*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x
+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+
b^2)^(1/2)*a^2-(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((
-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/
2))*(-a^2+b^2)^(1/2)*b^2-4*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+
e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2
*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e)
)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/
2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+2*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(
f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2)*(d*sin(f*x+e))^(5/2)*(g*cos(f*x+e))^(1/2)/cos(f*x+e)/sin(f*x+
e)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g \cos \left (f x + e\right )} \left (d \sin \left (f x + e\right )\right )^{\frac{5}{2}}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^(5/2)*(g*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate(sqrt(g*cos(f*x + e))*(d*sin(f*x + e))^(5/2)/(b*sin(f*x + e) + a), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^(5/2)*(g*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))**(5/2)*(g*cos(f*x+e))**(1/2)/(a+b*sin(f*x+e)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g \cos \left (f x + e\right )} \left (d \sin \left (f x + e\right )\right )^{\frac{5}{2}}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^(5/2)*(g*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(sqrt(g*cos(f*x + e))*(d*sin(f*x + e))^(5/2)/(b*sin(f*x + e) + a), x)