3.1324 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{3/2}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{c d \sqrt{b d+2 c d x}} \]

[Out]

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

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Rubi [A]  time = 0.726421, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{c d \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 129.484, size = 216, normalized size = 0.94 \[ - \frac{\sqrt{a + b x + c x^{2}}}{c d \sqrt{b d + 2 c d x}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c^{2} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{c^{2} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [C]  time = 0.75028, size = 176, normalized size = 0.77 \[ \frac{-c (b+2 c x) (a+x (b+c x))+\frac{i (b+2 c x)^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}}{c^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.06, size = 325, normalized size = 1.4 \[{\frac{1}{2\,{d}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ){c}^{2}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 4\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-{\it EllipticE} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ){b}^{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}-2\,{c}^{2}{x}^{2}-2\,bxc-2\,ac \right ) } \]

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)^(3/2),x)

[Out]

1/2*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(4*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))*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)-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))*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)-2*c^2*x^2-2*b*x*c-2*a*c)/d^2/
(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. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(d*(b + 2*c*x))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2), x)