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

Optimal. Leaf size=319 \[ -\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )} \]

[Out]

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

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Rubi [A]  time = 0.487925, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {944, 719, 419, 933, 168, 538, 537} \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 944

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

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)
])

Rubi steps

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

Mathematica [C]  time = 1.21592, size = 300, normalized size = 0.94 \[ -\frac{2 i \sqrt{f+g x} \sqrt{\frac{g \left (\sqrt{a}+i \sqrt{c} x\right )}{\sqrt{a} g-i \sqrt{c} f}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-\Pi \left (\frac{e \left (f-\frac{i \sqrt{a} g}{\sqrt{c}}\right )}{e f-d g};i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{g \left (\sqrt{c} x+i \sqrt{a}\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*Sqrt[f + g*x]*(EllipticF[I*ArcSinh[Sqrt
[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - El
lipticPi[(e*(f - (I*Sqrt[a]*g)/Sqrt[c]))/(e*f - d*g), I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt
[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(e*Sqrt[(Sqrt[c]*(f + g*x))/(g*(I*Sqrt[a] +
Sqrt[c]*x))]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.249, size = 439, normalized size = 1.4 \begin{align*} 2\,{\frac{\sqrt{gx+f}\sqrt{c{x}^{2}+a}}{ce \left ( cg{x}^{3}+cf{x}^{2}+agx+af \right ) }\sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{\sqrt{-ac}g-cf}}} \left ( f{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) c-\sqrt{-ac}{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) g-{\it EllipticPi} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},{\frac{ \left ( \sqrt{-ac}g-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) cf+{\it EllipticPi} \left ( \sqrt{-{\frac{c \left ( gx+f \right ) }{\sqrt{-ac}g-cf}}},{\frac{ \left ( \sqrt{-ac}g-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{\sqrt{-ac}g-cf}{\sqrt{-ac}g+cf}}} \right ) \sqrt{-ac}g \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-(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)*(f*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-(-a*c)^(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))*g-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*f+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)*g)/c/e/(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{\sqrt{g x + f}}{\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)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g x + f}}{\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)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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