3.2520 \(\int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=418 \[ \frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} (b+2 c x)}-\frac{\sqrt{2} \left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} (b+2 c x)}+\frac{2 (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )} \]

[Out]

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

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Rubi [A]  time = 0.708976, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{3/4} (b+2 c x)}-\frac{\sqrt{2} \left (b^2-4 a c\right )^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} (b+2 c x)}+\frac{2 (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 55.1698, size = 556, normalized size = 1.33 \[ \frac{2 \sqrt [4]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{\sqrt{c} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} - \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} + \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{2 c^{\frac{3}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(1/4),x)

[Out]

2*(a + b*x + c*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqr
t((b + 2*c*x)**2)/(sqrt(c)*(b + 2*c*x)*sqrt(-4*a*c + b**2)*(2*sqrt(c)*sqrt(a + b
*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) - sqrt(2)*sqrt(-(-4*a*c + b**2 + c*(4*a +
4*b*x + 4*c*x**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*
c + b**2) + 1)**2))*(-4*a*c + b**2)**(3/4)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqr
t(-4*a*c + b**2) + 1)*sqrt((b + 2*c*x)**2)*elliptic_e(2*atan(sqrt(2)*c**(1/4)*(a
+ b*x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(c**(3/4)*(b + 2*c*x)*sqrt
(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))) + sqrt(2)*sqrt(-(-4*a*c + b**2 + c
*(4*a + 4*b*x + 4*c*x**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqr
t(-4*a*c + b**2) + 1)**2))*(-4*a*c + b**2)**(3/4)*(2*sqrt(c)*sqrt(a + b*x + c*x*
*2)/sqrt(-4*a*c + b**2) + 1)*sqrt((b + 2*c*x)**2)*elliptic_f(2*atan(sqrt(2)*c**(
1/4)*(a + b*x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(2*c**(3/4)*(b + 2*
c*x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 0.10153, size = 126, normalized size = 0.3 \[ \frac{2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-1/4),x]

[Out]

(2^(3/4)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b
^2 - 4*a*c])^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*
c*x)/(2*Sqrt[b^2 - 4*a*c])])/(3*c*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.196, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(1/4),x)

[Out]

int(1/(c*x^2+b*x+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(-1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(-1/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral((a + b*x + c*x**2)**(-1/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(-1/4), x)