3.7.50 \(\int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx\) [650]

Optimal. Leaf size=746 \[ -\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} e \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} e f \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

-e^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(-d*g+e*f)/(e*x+d)-e*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)/(a
*e^2+c*d^2)/(-d*g+e*f)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)+e*f*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)/(a*e^2+c*d^2)/(-d*g+e*f)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)-
d*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)/(a*e^2+c*d^2)/(-d*g+e*f)/(g*x+f)^
(1/2)/(c*x^2+a)^(1/2)+(a*e^2*g-c*d*(-3*d*g+2*e*f))*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)/(a*e^2+c*d^2)/(-d*g+e*f)/(e+d*c^(1/2)/(-a)^(1/2))/(g*x+f)^(1/2)/(c*x^2+a)^
(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.35, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {954, 6874, 733, 430, 858, 435, 947, 174, 552, 551} \begin {gather*} \frac {\sqrt {-a} \sqrt {c} e f \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} 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 )}{\sqrt {a+c x^2} \sqrt {f+g x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac {\sqrt {-a} \sqrt {c} d g \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} 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 )}{\sqrt {a+c x^2} \sqrt {f+g x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac {\sqrt {-a} \sqrt {c} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} 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 )}{\sqrt {a+c x^2} \left (a e^2+c d^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {\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-c d (2 e f-3 d g)\right ) \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 )}{\sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right ) (e f-d g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x))) - (Sqrt[-a]*Sqrt[c]*e*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)])/((c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) +
(Sqrt[-a]*Sqrt[c]*e*f*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)])/((c*d^2 + a*e^2)*(e*f - d*g)*Sq
rt[f + g*x]*Sqrt[a + c*x^2]) - (Sqrt[-a]*Sqrt[c]*d*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]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/
((c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2]) + ((a*e^2*g - c*d*(2*e*f - 3*d*g))*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[Sq
rt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(((Sqrt[c]*d)/Sqrt[-a] + e)*(
c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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 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 954

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d +
e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 + a*e^2))), x] + Dist[1/(2*(m + 1)*(e
*f - d*g)*(c*d^2 + a*e^2)), Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[2*d*(c*e*f - c*d*g)*(
m + 1) - a*e^2*g*(2*m + 3) + 2*e*(c*d*g*(m + 1) - c*e*f*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; Free
Q[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\int \frac {a e^2 g-2 c d (e f-d g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\int \left (-\frac {c d g}{\sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c e g x}{\sqrt {f+g x} \sqrt {a+c x^2}}+\frac {a e^2 g-c d (2 e f-3 d g)}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}+\frac {(c d g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}+\frac {(c e g) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}+\frac {(c e) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {(c e f) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right ) \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}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} d 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 )}{\sqrt {-a} \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} d 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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} e \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} \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (a \sqrt {c} e f \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 )}{\sqrt {-a} \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} e \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} e f \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d 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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} e \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} e f \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d 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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \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 ) \left (c d^2+a e^2\right ) (e f-d g) \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 = 24.55, size = 1349, normalized size = 1.81 \begin {gather*} \frac {\sqrt {f+g x} \left (-\frac {2 e^2 \left (a+c x^2\right )}{d+e x}+\frac {2 \left (-c e^2 f^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+c d e f^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-a e^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+2 c e^2 f^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-2 c d e f g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-c e^2 f \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+\sqrt {c} e \left (-i \sqrt {c} f+\sqrt {a} g\right ) (-e f+d g) \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} d-i \sqrt {a} e\right ) g \left (\sqrt {a} e g+i \sqrt {c} (e f-2 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} 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 )-2 i c d e f g \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 c d^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 )+i a e^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 )\right )}{g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (-e f+d g) (f+g x)}\right )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[f + g*x]*((-2*e^2*(a + c*x^2))/(d + e*x) + (2*(-(c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + c*d*e*f^2
*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*d*e*g^3*Sqrt[-f - (I*Sq
rt[a]*g)/Sqrt[c]] + 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*
g)/Sqrt[c]]*(f + g*x) - c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)
/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + 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]*d - I*Sq
rt[a]*e)*g*(Sqrt[a]*e*g + I*Sqrt[c]*(e*f - 2*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sq
rt[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)] - (2*I)*c*d*e*f*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)*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)*c*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)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*S
qrt[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)] + I*a*e^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)]))/(g*Sqrt[-f -
(I*Sqrt[a]*g)/Sqrt[c]]*(-(e*f) + d*g)*(f + g*x))))/(2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5737\) vs. \(2(627)=1254\).
time = 0.10, size = 5738, normalized size = 7.69

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{\left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \left (e x +d \right )}-\frac {c d g \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 )}{\left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {c e g \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 )}{\left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {\left (a \,e^{2} g +3 c \,d^{2} g -2 c d e f \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}}}\, \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 (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) e \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}}\)	\(995\)
default	\(\text {Expression too large to display}\)	\(5738\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^2 + a)*sqrt(g*x + f)*(x*e + d)^2), 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)^2/(g*x+f)^(1/2)/(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 )^{2} \sqrt {f + g x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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