3.643 \(\int \frac{(f+g x)^{5/2}}{(d+e x) \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ \frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g^2+c \left (-3 d^2 g^2+6 d e f g-2 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right ),\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right )}{3 c^{3/2} e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (7 e f-3 d g) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right )|\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right )}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 (e f-d g)^2 \sqrt{\frac{g \left (\sqrt{-a}-\sqrt{c} x\right )}{\sqrt{-a} g+\sqrt{c} f}} \sqrt{-\frac{g \left (\sqrt{-a}+\sqrt{c} x\right )}{\sqrt{c} f-\sqrt{-a} g}} \Pi \left (\frac{e \left (f+\frac{\sqrt{-a} g}{\sqrt{c}}\right )}{e f-d g};\sin ^{-1}\left (\sqrt{\frac{c}{c f+\sqrt{-a} \sqrt{c} g}} \sqrt{f+g x}\right )|\frac{\sqrt{c} f+\sqrt{-a} g}{\sqrt{c} f-\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{\frac{c}{\sqrt{-a} \sqrt{c} g+c f}}}+\frac{2 g^2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 c e} \]

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) - (2*Sqrt[-a]*g*(7*e*f - 3*d*g)*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a
]*EllipticE[ArcSin[Sqrt[1 + (a*Sqrt[c]*x)/(-a)^(3/2)]/Sqrt[2]], (2*a*g)/(-(Sqrt[-a]*Sqrt[c]*f) + a*g)])/(3*Sqr
t[c]*e^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*g*(a*e^2*g^2 + c*(-
2*e^2*f^2 + 6*d*e*f*g - 3*d^2*g^2))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ell
ipticF[ArcSin[Sqrt[1 + (a*Sqrt[c]*x)/(-a)^(3/2)]/Sqrt[2]], (2*a*g)/(-(Sqrt[-a]*Sqrt[c]*f) + a*g)])/(3*c^(3/2)*
e^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)^2*Sqrt[(g*(Sqrt[-a] - Sqrt[c]*x))/(Sqrt[c]*f + Sqrt[-a]*g)
]*Sqrt[-((g*(Sqrt[-a] + Sqrt[c]*x))/(Sqrt[c]*f - Sqrt[-a]*g))]*EllipticPi[(e*(f + (Sqrt[-a]*g)/Sqrt[c]))/(e*f
- d*g), ArcSin[Sqrt[c/(c*f + Sqrt[-a]*Sqrt[c]*g)]*Sqrt[f + g*x]], (Sqrt[c]*f + Sqrt[-a]*g)/(Sqrt[c]*f - Sqrt[-
a]*g)])/(e^3*Sqrt[c/(c*f + Sqrt[-a]*Sqrt[c]*g)]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.946193, antiderivative size = 808, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {958, 719, 419, 933, 168, 538, 537, 424, 743, 844} \[ -\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right ) (e f-d g)^3}{e^3 \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) (e f-d g)^2}{\sqrt{c} e^3 \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) (e f-d g)}{\sqrt{c} e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{8 \sqrt{-a} f g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 \sqrt{c} e \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{2 \sqrt{-a} g \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 c^{3/2} e \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 g^2 \sqrt{f+g x} \sqrt{c x^2+a}}{3 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) - (8*Sqrt[-a]*f*g*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[Ar
cSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*Sqrt[c]*e*Sqrt[(Sqrt[c]
*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[1 + (c*x
^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(Sqrt[c
]*e^2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (2*Sqrt[-a]*g*(e*f - d*g)^2*Sqrt[(
Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a
]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(Sqrt[c]*e^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*g
*(c*f^2 + a*g^2)*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[
1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3*c^(3/2)*e*Sqrt[f + g*x]*Sqrt[a +
c*x^2]) - (2*(e*f - d*g)^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticPi[(
2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + S
qrt[-a]*g)])/(e^3*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqrt[f + g*x]*Sqrt[a + c*x^2])

Rule 958

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 743

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

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

Mathematica [C]  time = 9.72578, size = 1440, normalized size = 2.4 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} g^2}{3 c e}+\frac{2 (f+g x)^{3/2} \left (\frac{7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3}{(f+g x)^2}+\frac{3 i c e^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right ) f^2}{\sqrt{f+g x}}-\frac{14 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{f+g x}-\frac{3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{(f+g x)^2}-\frac{6 i c d e g \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right ) f}{\sqrt{f+g x}}+7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+\frac{6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{f+g x}+\frac{7 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{(f+g x)^2}+\frac{\sqrt{c} e \left (\sqrt{a} g-i \sqrt{c} f\right ) (7 e f-3 d g) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{f+g x}}+\frac{i e \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (i \sqrt{a} e g+\sqrt{c} (6 e f-3 d g)\right ) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{f+g x}}+\frac{3 i c d^2 g^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )}{\sqrt{f+g x}}-3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\frac{3 a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{(f+g x)^2}\right )}{3 c e^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{c (f+g x)^2 \left (\frac{f}{f+g x}-1\right )^2}{g^2}+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (2*(f + g*x)^(3/2)*(7*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]
- 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (7*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (
3*c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (7*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])
/(f + g*x)^2 - (3*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (14*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]
*g)/Sqrt[c]])/(f + g*x) + (6*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*
f + Sqrt[a]*g)*(7*e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x)
+ (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqr
t[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (I*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(I*Sqrt[a]*e*
g + Sqrt[c]*(6*e*f - 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) +
(I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[
c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*e^2*f^2*Sqrt[1 - f/(f + g*x) - (I*Sqr
t[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*
f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f
- I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - ((6*I)*c*d*e*f*g*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]
*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f -
d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I
*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/
(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g)
)/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqr
t[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(3*c*e^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f +
g*x)^2*(-1 + f/(f + g*x))^2)/g^2])

________________________________________________________________________________________

Maple [B]  time = 0.273, size = 3164, normalized size = 5.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(
1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f
))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c*d*e*g^3-6*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(
1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*Elli
pticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c*e^2*f*g^
2-(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2
))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a
*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*a*e^2*g^3-3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))
*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)
^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^2*d^2*f*g^2+9*(-(g*x+f)*c/((-a*c)^(
1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*
f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2)
)*c^2*d*e*f^2*g-9*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*(
(c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(
1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^2*e^2*f^3+3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(
1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((
-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c*d^2*g^3-6*(-(g*x+
f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a
*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)
*g+c*f))^(1/2))*(-a*c)^(1/2)*c*d*e*f*g^2+2*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a
*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*
g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c*e^2*f^2*g-3*(-(g*x+f)*c/((-a*
c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*
g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(
1/2))*a*c*d*e*g^3+7*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)
*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)
^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c*e^2*f*g^2-3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*
c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*
c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^2*d*e*f^2*g+7*(-(g*x+f)*c/
((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(
1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*
f))^(1/2))*c^2*e^2*f^3+3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^
(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),((-
a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^2*d^2*f*g^2-6*(-(g*x+f)*
c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)
^(1/2)*g-c*f))^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-(
(-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^2*d*e*f^2*g+3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*
x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticPi((-(
g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g
+c*f))^(1/2))*c^2*e^2*f^3-3*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f
))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),
((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c*d^2*g^3+
6*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2
))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),((-a*c)^(1/2)*g-c*f)*e/c/(
d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c*d*e*f*g^2-3*(-(g*x+f)*c/((-a*c)^(1
/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f
))^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)
*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c*e^2*f^2*g-x^3*c^2*e^2*g^3-x^2*c^2*e^2*f*g^2-x*a*c*e^2*g^3-
a*c*e^2*f*g^2)/e^3/c^2/(c*g*x^3+c*f*x^2+a*g*x+a*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f + g*x)**(5/2)/(sqrt(a + c*x**2)*(d + e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError