3.7.53 \(\int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx\) [653]

Optimal. Leaf size=818 \[ \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} e g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

2/3*g^2*(c*x^2+a)^(1/2)/(-d*g+e*f)/(a*g^2+c*f^2)/(g*x+f)^(3/2)+8/3*c*f*g^2*(c*x^2+a)^(1/2)/(-d*g+e*f)/(a*g^2+c
*f^2)^2/(g*x+f)^(1/2)+2*e*g^2*(c*x^2+a)^(1/2)/(-d*g+e*f)^2/(a*g^2+c*f^2)/(g*x+f)^(1/2)+8/3*c^(3/2)*f*g*Ellipti
cE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(
1/2)*(c*x^2/a+1)^(1/2)/(-d*g+e*f)/(a*g^2+c*f^2)^2/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(
1/2)+2*e*g*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-
a)^(1/2)*c^(1/2)*(g*x+f)^(1/2)*(c*x^2/a+1)^(1/2)/(-d*g+e*f)^2/(a*g^2+c*f^2)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(
g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2/3*g*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)
^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(c*x^2/a+1)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/
(-d*g+e*f)/(a*g^2+c*f^2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-2*e^2*EllipticPi(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(
1/2),2*e/(e+d*c^(1/2)/(-a)^(1/2)),2^(1/2)*(g*(-a)^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2))*(c*x^2/a+1)^(1/2)*((g
*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/(-d*g+e*f)^2/(e+d*c^(1/2)/(-a)^(1/2))/(g*x+f)^(1/2)/(c*x^2+a)^(1
/2)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {972, 759, 849, 858, 733, 435, 430, 21, 947, 174, 552, 551} \begin {gather*} -\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};\text {ArcSin}\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^2}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\text {ArcSin}\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}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\text {ArcSin}\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 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\text {ArcSin}\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 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {8 c f g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(f + g*x)^(5/2)*Sqrt[a + c*x^2]),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 174

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 430

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

Rule 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 733

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 759

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*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] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 858

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]

Rule 947

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 972

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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx &=\int \left (-\frac {g}{(e f-d g) (f+g x)^{5/2} \sqrt {a+c x^2}}-\frac {e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt {a+c x^2}}+\frac {e^2}{(e f-d g)^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {e^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {1}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {g \int \frac {1}{(f+g x)^{5/2} \sqrt {a+c x^2}} \, dx}{e f-d g}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {(2 c e g) \int \frac {-\frac {f}{2}-\frac {g x}{2}}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}+\frac {(2 c g) \int \frac {-\frac {3 f}{2}+\frac {g x}{2}}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}+\frac {\left (e^2 \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 f-d g)^2 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {(4 c g) \int \frac {\frac {1}{4} \left (3 c f^2-a g^2\right )+c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}-\frac {(c e g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \sqrt {1+\frac {c x^2}{a}}\right ) \text {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 f-d g)^2 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {\left (4 c^2 f g\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}+\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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 f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} e g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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} (e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (8 a c^{3/2} f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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} (e f-d g) \left (c f^2+a g^2\right )^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 \sqrt {c} g \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {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} (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} 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 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \sqrt {c} g \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 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 25.90, size = 1917, normalized size = 2.34 \begin {gather*} \frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g) \left (a+c x^2\right ) \left (a g^2 (4 e f-d g+3 e g x)+c f (-d g (5 f+4 g x)+e f (8 f+7 g x))\right )-(f+g x) \left (7 c^2 e^2 f^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-11 c^2 d e f^4 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+4 c^2 d^2 f^3 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+10 a c e^2 f^3 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-14 a c d e f^2 g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+4 a c d^2 f g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+3 a^2 e^2 f g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-3 a^2 d e g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-14 c^2 e^2 f^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+22 c^2 d e f^3 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-8 c^2 d^2 f^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-6 a c e^2 f^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+6 a c d e f g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+7 c^2 e^2 f^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2-11 c^2 d e f^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+4 c^2 d^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+3 a c e^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2-3 a c d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+\sqrt {c} \left (-i \sqrt {c} f+\sqrt {a} g\right ) (e f-d g) \left (3 a e g^2+c f (7 e f-4 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} 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 )+\left (\sqrt {c} f+i \sqrt {a} g\right ) \left (3 a^{3/2} e^2 g^3+3 i a \sqrt {c} e g^2 (2 e f-d g)+\sqrt {a} c g \left (2 e^2 f^2+2 d e f g-d^2 g^2\right )+3 i c^{3/2} f \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} F\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 )-3 i c^2 e^2 f^4 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )-6 i a c e^2 f^2 g^2 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )-3 i a^2 e^2 g^4 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )\right )\right )}{3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g)^3 \left (c f^2+a g^2\right )^2 (f+g x)^{3/2} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(f + g*x)^(5/2)*Sqrt[a + c*x^2]),x]

[Out]

(2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*(a + c*x^2)*(a*g^2*(4*e*f - d*g + 3*e*g*x) + c*f*(-(d*g*(
5*f + 4*g*x)) + e*f*(8*f + 7*g*x))) - (f + g*x)*(7*c^2*e^2*f^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 11*c^2*d*e*f
^4*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 4*c^2*d^2*f^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 10*a*c*e^2*f^3*g^
2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 14*a*c*d*e*f^2*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 4*a*c*d^2*f*g^4*Sqr
t[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 3*a^2*e^2*f*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a^2*d*e*g^5*Sqrt[-f - (I*
Sqrt[a]*g)/Sqrt[c]] - 14*c^2*e^2*f^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 22*c^2*d*e*f^3*g*Sqrt[-f - (
I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 8*c^2*d^2*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 6*a*c*e^2*f^2
*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 6*a*c*d*e*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) +
7*c^2*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - 11*c^2*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]
]*(f + g*x)^2 + 4*c^2*d^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 3*a*c*e^2*f*g^2*Sqrt[-f - (I*Sq
rt[a]*g)/Sqrt[c]]*(f + g*x)^2 - 3*a*c*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*((-I)*Sqr
t[c]*f + Sqrt[a]*g)*(e*f - d*g)*(3*a*e*g^2 + c*f*(7*e*f - 4*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)
]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[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[c]*f + I*Sqrt[a]*g)*(3*
a^(3/2)*e^2*g^3 + (3*I)*a*Sqrt[c]*e*g^2*(2*e*f - d*g) + Sqrt[a]*c*g*(2*e^2*f^2 + 2*d*e*f*g - d^2*g^2) + (3*I)*
c^(3/2)*f*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*
g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)] - (3*I)*c^2*e^2*f^4*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x
))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)] - (6*I)*a*c*e^2*f^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[
-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)] - (3*I)*a^2*e^2*g^4*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c
] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)])))/(3*S
qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^3*(c*f^2 + a*g^2)^2*(f + g*x)^(3/2)*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(9414\) vs. \(2(683)=1366\).
time = 0.13, size = 9415, normalized size = 11.51

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 \left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \left (x +\frac {f}{g}\right )^{2}}+\frac {2 \left (c g \,x^{2}+a g \right ) g \left (3 a e \,g^{2}-4 c d f g +7 c e \,f^{2}\right )}{3 \left (a \,g^{2}+c \,f^{2}\right )^{2} \left (d g -e f \right )^{2} \sqrt {\left (x +\frac {f}{g}\right ) \left (c g \,x^{2}+a g \right )}}+\frac {2 \left (-\frac {c g}{3 \left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right )}-\frac {c f g \left (3 a e \,g^{2}-4 c d f g +7 c e \,f^{2}\right )}{3 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 g^{2} c \left (3 a e \,g^{2}-4 c d f g +7 c e \,f^{2}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 e \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\left (d g -e f \right )^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(1079\)
default	\(\text {Expression too large to display}\)	\(9415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (f+g\,x\right )}^{5/2}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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