3.648 \(\int \frac{1}{\left (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=444 \[ -\frac{\left (176-23 \left (1-\frac{6}{x}\right )^2\right ) x^2}{51759 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{\left (45401-3722 \left (1-\frac{6}{x}\right )^2\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \left (\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}-\frac{\left (7444-145 \sqrt{613}\right ) \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{207036\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{51759\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

[Out]

-((176 - 23*(1 - 6/x)^2)*x^2)/(51759*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) +
((45401 - 3722*(1 - 6/x)^2)*(1 - 6/x)*x^2)/(31728267*Sqrt[9 - 6*x - 44*x^2 + 15*
x^3 + 3*x^4]) + (3722*(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)*(1 - 6/x)*x^2)/(317
28267*(Sqrt[613] + (6 - x)^2/x^2)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) + (37
22*Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*(S
qrt[613] + (6 - x)^2/x^2)*x^2*EllipticE[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 +
91*Sqrt[613])/1226])/(51759*613^(3/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) -
((7444 - 145*Sqrt[613])*Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613]
+ (6 - x)^2/x^2)^2]*(Sqrt[613] + (6 - x)^2/x^2)*x^2*EllipticF[2*ArcTan[(6 - x)/(
613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(207036*613^(3/4)*Sqrt[9 - 6*x - 44*x
^2 + 15*x^3 + 3*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.887108, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\left (176-23 \left (1-\frac{6}{x}\right )^2\right ) x^2}{51759 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{\left (45401-3722 \left (1-\frac{6}{x}\right )^2\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \left (\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}-\frac{\left (7444-145 \sqrt{613}\right ) \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{207036\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{51759\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]

[Out]

-((176 - 23*(1 - 6/x)^2)*x^2)/(51759*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) +
((45401 - 3722*(1 - 6/x)^2)*(1 - 6/x)*x^2)/(31728267*Sqrt[9 - 6*x - 44*x^2 + 15*
x^3 + 3*x^4]) + (3722*(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)*(1 - 6/x)*x^2)/(317
28267*(Sqrt[613] + (6 - x)^2/x^2)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) + (37
22*Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*(S
qrt[613] + (6 - x)^2/x^2)*x^2*EllipticE[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 +
91*Sqrt[613])/1226])/(51759*613^(3/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) -
((7444 - 145*Sqrt[613])*Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613]
+ (6 - x)^2/x^2)^2]*(Sqrt[613] + (6 - x)^2/x^2)*x^2*EllipticF[2*ArcTan[(6 - x)/(
613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(207036*613^(3/4)*Sqrt[9 - 6*x - 44*x
^2 + 15*x^3 + 3*x^4])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 1296 \int ^{- \frac{1}{6} + \frac{1}{x}} \frac{1}{\left (\frac{15116544 x^{4} - 76422528 x^{2} + 7150032}{\left (- 36 x - 6\right )^{4}}\right )^{\frac{3}{2}} \left (- 36 x - 6\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(3/2),x)

[Out]

-1296*Integral(1/(((15116544*x**4 - 76422528*x**2 + 7150032)/(-36*x - 6)**4)**(3
/2)*(-36*x - 6)**2), (x, -1/6 + 1/x))

_______________________________________________________________________________________

Mathematica [C]  time = 6.06871, size = 5428, normalized size = 12.23 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [C]  time = 0.025, size = 5427, normalized size = 12.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2),x, algorithm="maxima")

[Out]

integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2),x, algorithm="fricas")

[Out]

integral((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(3/2),x)

[Out]

Integral((3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9)**(-3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2),x, algorithm="giac")

[Out]

integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)