\(\int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx\) [652]

Optimal result

Integrand size = 28, antiderivative size = 387 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {a+c x^2}}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

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

Time = 0.49 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {972, 759, 21, 733, 435, 947, 174, 552, 551} \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {a+c x^2} \left (a g^2+c f^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {2 e \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)}+\frac {2 g^2 \sqrt {a+c x^2}}{\sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g)} \]

[In]

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

[Out]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 i \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \left (\sqrt {c} (e f-d g) E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\left (i \sqrt {a} e g+\sqrt {c} (-2 e f+d g)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+e \left (\sqrt {c} f-i \sqrt {a} g\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g)^2 \sqrt {a+c x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(928\) vs. \(2(324)=648\).

Time = 2.53 (sec) , antiderivative size = 929, normalized size of antiderivative = 2.40

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c g \,x^{2}+a g \right ) g}{\left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {\left (x +\frac {f}{g}\right ) \left (c g \,x^{2}+a g \right )}}+\frac {2 c g f \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}}}\, F\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 \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 c \,g^{2} \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 ) E\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}\, F\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 \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 \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}}}\, \Pi \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 ) \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}}\)	\(929\)
default	\(\text {Expression too large to display}\)	\(2011\)

[In]

int(1/(e*x+d)/(g*x+f)^(3/2)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(g*x+f)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)/(g*x+f)**(3/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**2)*(d + e*x)*(f + g*x)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)/(g*x+f)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)/(g*x+f)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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