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

Optimal. Leaf size=288 \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} g \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} g \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {844, 719, 424, 419} \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} g \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} g \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [C]  time = 1.68, size = 439, normalized size = 1.52 \[ -\frac {2 \left (\sqrt {c} g (f+g x)^{3/2} \left (\sqrt {a} e-i \sqrt {c} d\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-e g^2 \left (a+c x^2\right ) \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+i \sqrt {c} e (f+g x)^{3/2} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i \sqrt {a} g}{\sqrt {c}}}{f+g x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^2 \sqrt {a+c x^2} \sqrt {f+g x} \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}}{c g x^{3} + c f x^{2} + a g x + a f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e x + d}{\sqrt {c x^{2} + a} \sqrt {g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.04, size = 520, normalized size = 1.81 \[ \frac {2 \left (-a e \,g^{2} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+a e \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+c d f g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c e \,f^{2} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, d \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+\sqrt {-a c}\, e f g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{-c f +\sqrt {-a c}\, g}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{c f +\sqrt {-a c}\, g}}\, \sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{\left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right ) c \,g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e x + d}{\sqrt {c x^{2} + a} \sqrt {g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {d+e\,x}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________