3.1640 \(\int \frac{b+2 c x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=391 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e \sqrt{d+e x} \sqrt{a+b x+c x^2}} \]

[Out]

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4
*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]
/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e*Sqr
t[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2
*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[S
qrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 -
4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 0.674183, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e \sqrt{d+e x} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4
*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]
/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e*Sqr
t[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2
*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[S
qrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 -
4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi in Sympy [A]  time = 115.075, size = 369, normalized size = 0.94 \[ \frac{2 \sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{e \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}}} + \frac{2 \sqrt{2} \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{c e \sqrt{d + e x} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)/(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

2*sqrt(2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*sqrt(d + e*x)*sqrt(-4*a*c +
b**2)*elliptic_e(asin(sqrt(2)*sqrt((b + 2*c*x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c
+ b**2))/2), 2*e*sqrt(-4*a*c + b**2)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(e*
sqrt(c*(-d - e*x)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))*sqrt(a + b*x + c*x**2))
+ 2*sqrt(2)*sqrt(c*(-d - e*x)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))*sqrt(c*(a
+ b*x + c*x**2)/(4*a*c - b**2))*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*elliptic_f(asi
n(sqrt(2)*sqrt((b + 2*c*x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2))/2), 2*e*sq
rt(-4*a*c + b**2)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(c*e*sqrt(d + e*x)*sqrt
(a + b*x + c*x**2))

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Mathematica [C]  time = 3.9649, size = 793, normalized size = 2.03 \[ \frac{(d+e x)^{3/2} \left (\frac{4 e^2 (a+x (b+c x)) \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}{(d+e x)^2}+\frac{i \sqrt{2} \sqrt{e^2 \left (b^2-4 a c\right )} \sqrt{\frac{(d+e x) \sqrt{e^2 \left (b^2-4 a c\right )}-2 a e^2+b e (d-e x)+2 c d e x}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{(d+e x) \sqrt{e^2 \left (b^2-4 a c\right )}+2 a e^2+b e (e x-d)-2 c d e x}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) \sqrt{\frac{(d+e x) \sqrt{e^2 \left (b^2-4 a c\right )}-2 a e^2+b e (d-e x)+2 c d e x}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{(d+e x) \sqrt{e^2 \left (b^2-4 a c\right )}+2 a e^2+b e (e x-d)-2 c d e x}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}\right )}{e^2 \sqrt{a+x (b+c x)} \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(3/2)*((4*e^2*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b
^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))/(d + e*x)^2 - (I*Sqrt[2]*(2*c*d - b*e + Sq
rt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + 2*c*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 -
4*a*c)*e^2]*(d + e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt
[(2*a*e^2 - 2*c*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2
*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*S
qrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d +
e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a
*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*Sqrt[(b^2 - 4*a*c)*e^2]*Sqrt[(-2*a*e^2 +
2*c*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*c*d - b*e +
Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 - 2*c*d*e*x + b*e*(-d + e*x)
+ Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(
d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b
*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*
a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(e^2*Sqrt[
(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + x*
(b + c*x)])

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Maple [B]  time = 0.056, size = 837, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

(4*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-
4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c*e^2-Ell
ipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+
b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*e^2-EllipticF
(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(
1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*b*e^
2+2*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(
-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^
2)^(1/2)*c*d*e-4*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))
^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2
))*a*c*e^2+4*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/
2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b
*c*d*e-4*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(
-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*c^2*d
^2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(
-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*2^(1/2)*(-(
e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/
2)/e^2/c/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, c x + b}{\sqrt{c x^{2} + b x + a} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{2 \, c x + b}{\sqrt{c x^{2} + b x + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

integral((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b + 2 c x}{\sqrt{d + e x} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)/(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out