3.1318 \(\int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx\)

Optimal. Leaf size=137 \[ \frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{3 c d}-\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 c^2 \sqrt{d} \sqrt{a+b x+c x^2}} \]

[Out]

(Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(3*c*d) - ((b^2 - 4*a*c)^(5/4)*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])/(3*c^2*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.340814, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{3 c d}-\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 c^2 \sqrt{d} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(3*c*d) - ((b^2 - 4*a*c)^(5/4)*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])/(3*c^2*Sqrt[d]*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 74.9296, size = 126, normalized size = 0.92 \[ \frac{\sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{3 c d} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 c^{2} \sqrt{d} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(3*c*d) - sqrt(c*(a + b*x + c*x**2)/(
4*a*c - b**2))*(-4*a*c + b**2)**(5/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(
d)*(-4*a*c + b**2)**(1/4))), -1)/(3*c**2*sqrt(d)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.315946, size = 149, normalized size = 1.09 \[ \frac{c (b+2 c x) (a+x (b+c x))-\frac{i \left (b^2-4 a c\right ) (b+2 c x)^{3/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{3 c^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.043, size = 364, normalized size = 2.7 \[{\frac{1}{6\,d \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ){c}^{2}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 4\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac-\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{2}+4\,{c}^{3}{x}^{3}+6\,b{c}^{2}{x}^{2}+4\,a{c}^{2}x+2\,x{b}^{2}c+2\,abc \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)/d*(4*((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)*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))*(-4*a*c+b^2)^(1/2)*a*c-((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)*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))*(-4*a*c+b^2)
^(1/2)*b^2+4*c^3*x^3+6*b*c^2*x^2+4*a*c^2*x+2*x*b^2*c+2*a*b*c)/(2*c^2*x^3+3*b*c*x
^2+2*a*c*x+b^2*x+a*b)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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