3.2527 \(\int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=170 \[ \frac{\sqrt{2} \sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{c} (b+2 c x)} \]

[Out]

(Sqrt[2]*(b^2 - 4*a*c)^(1/4)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*S
qrt[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])*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])/(c^(1/4)*(b + 2*c*x))

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Rubi [A]  time = 0.268518, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{2} \sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{c} (b+2 c x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-3/4),x]

[Out]

(Sqrt[2]*(b^2 - 4*a*c)^(1/4)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*S
qrt[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])*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])/(c^(1/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 15.0524, size = 219, normalized size = 1.29 \[ \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \sqrt [4]{- 4 a c + b^{2}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{\sqrt [4]{c} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(3/4),x)

[Out]

sqrt(2)*sqrt(-(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))/((4*a*c - b**2)*(2*sq
rt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)**2))*(-4*a*c + b**2)**(1/4
)*(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)/(c**(1/4)*(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 = 0.132527, size = 123, normalized size = 0.72 \[ \frac{\sqrt [4]{2} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-3/4),x]

[Out]

(2^(1/4)*(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])^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (-b + Sqrt[b^2 - 4*a*c] - 2*
c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 0.193, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(3/4),x)

[Out]

int(1/(c*x^2+b*x+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(-3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(-3/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**(3/4),x)

[Out]

Integral((a + b*x + c*x**2)**(-3/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-3/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(-3/4), x)