3.1359 \(\int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{2 \sqrt{d} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{a+b x+c x^2}} \]

[Out]

(2*(b^2 - 4*a*c)^(3/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(c*Sqrt[a
+ b*x + c*x^2]) - (2*(b^2 - 4*a*c)^(3/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b
^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
], -1])/(c*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.630096, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{d} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 4*a*c)^(3/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(c*Sqrt[a
+ b*x + c*x^2]) - (2*(b^2 - 4*a*c)^(3/4)*Sqrt[d]*Sqrt[-((c*(a + b*x + c*x^2))/(b
^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
], -1])/(c*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 109.129, size = 185, normalized size = 0.95 \[ \frac{2 \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c \sqrt{a + b x + c x^{2}}} - \frac{2 \sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(3/4)*ellip
tic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(c*sqrt(a
+ b*x + c*x**2)) - 2*sqrt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c +
b**2)**(3/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4
))), -1)/(c*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.270143, size = 151, normalized size = 0.77 \[ \frac{2 i \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c d \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(Ellipti
cE[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[
Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(c*d*(-((b + 2*c*x)/Sqrt[b^2 - 4*
a*c]))^(3/2)*Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.021, size = 186, normalized size = 1. \[{\frac{4\,ac-{b}^{2}}{c \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticE} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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