3.2534 \(\int \frac{1}{\left (a+b x+c x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=451 \[ -\frac{4 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{8 \sqrt{c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{2 \sqrt{2} \sqrt [4]{c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{4 \sqrt{2} \sqrt [4]{c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)} \]

[Out]

(-4*(b + 2*c*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/4)) + (8*Sqrt[c]*(b + 2*c*x
)*(a + b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c
*x^2])/Sqrt[b^2 - 4*a*c])) - (4*Sqrt[2]*c^(1/4)*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]*
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])/((b^2 - 4*a*c)^(1/4)*(b + 2*c
*x)) + (2*Sqrt[2]*c^(1/4)*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]*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])/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.79368, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{4 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{8 \sqrt{c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{2 \sqrt{2} \sqrt [4]{c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{4 \sqrt{2} \sqrt [4]{c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-5/4),x]

[Out]

(-4*(b + 2*c*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/4)) + (8*Sqrt[c]*(b + 2*c*x
)*(a + b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c
*x^2])/Sqrt[b^2 - 4*a*c])) - (4*Sqrt[2]*c^(1/4)*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]*
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])/((b^2 - 4*a*c)^(1/4)*(b + 2*c
*x)) + (2*Sqrt[2]*c^(1/4)*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]*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])/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 71.0114, size = 588, normalized size = 1.3 \[ - \frac{4 \sqrt{2} \sqrt [4]{c} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{\left (b + 2 c x\right ) \sqrt [4]{- 4 a c + b^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} + \frac{2 \sqrt{2} \sqrt [4]{c} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{\left (b + 2 c x\right ) \sqrt [4]{- 4 a c + b^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} + \frac{8 \sqrt{c} \sqrt [4]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} - \frac{4 \left (b + 2 c x\right )}{\left (- 4 a c + b^{2}\right ) \sqrt [4]{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(5/4),x)

[Out]

-4*sqrt(2)*c**(1/4)*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))*(2*sqrt(c
)*sqrt(a + b*x + c*x**2)/sqrt(-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
)/((b + 2*c*x)*(-4*a*c + b**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*
x**2))) + 2*sqrt(2)*c**(1/4)*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))*
(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)/((b + 2*c*x)*(-4*a*c + b**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b
*x + 4*c*x**2))) + 8*sqrt(c)*(a + b*x + c*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4
*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/((b + 2*c*x)*(-4*a*c + b**2)**(3/2)
*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) - 4*(b + 2*c*x)/((-
4*a*c + b**2)*(a + b*x + c*x**2)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.353037, size = 161, normalized size = 0.36 \[ \frac{4 \left (2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{2 c x \sqrt{b^2-4 a c}+b \sqrt{b^2-4 a c}-4 a c+b^2}{b^2-4 a c}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )-3 (b+2 c x)\right )}{3 \left (b^2-4 a c\right ) \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-5/4),x]

[Out]

(4*(-3*(b + 2*c*x) + 2^(3/4)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b^2 - 4*a*c + b*S
qrt[b^2 - 4*a*c] + 2*c*Sqrt[b^2 - 4*a*c]*x)/(b^2 - 4*a*c))^(1/4)*Hypergeometric2
F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(3*(
b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.205, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^(5/4),x)

[Out]

int(1/(c*x^2+b*x+a)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-5/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(-5/4), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-5/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(-5/4), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral((a + b*x + c*x**2)**(-5/4), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-5/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(-5/4), x)