3.1354 \(\int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{d} \sqrt{a+b x+c x^2}} \]

[Out]

(2*(b^2 - 4*a*c)^(1/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Ar
cSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(c*Sqrt[d]*Sqrt[a
+ b*x + c*x^2])

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Rubi [A]  time = 0.262011, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt{d} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 4*a*c)^(1/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Ar
cSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(c*Sqrt[d]*Sqrt[a
+ b*x + c*x^2])

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Rubi in Sympy [A]  time = 68.2988, size = 92, normalized size = 0.95 \[ \frac{2 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt [4]{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c \sqrt{d} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(1/4)*elliptic_f(as
in(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(c*sqrt(d)*sqrt(a
+ b*x + c*x**2))

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Mathematica [C]  time = 0.253533, size = 116, normalized size = 1.2 \[ \frac{2 i \sqrt{a+x (b+c x)} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} \sqrt{b+2 c x} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 189, normalized size = 2. \[{\frac{1}{cd \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

EllipticF(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))*((-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)*(-
4*a*c+b^2)^(1/2)*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)/c/d/(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. \[ \int \frac{1}{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

integral(1/(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. \[ \int \frac{1}{\sqrt{d \left (b + 2 c x\right )} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(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. \[ \int \frac{1}{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)), x)