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

Optimal. Leaf size=209 \[ -\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*x*Sqrt[a*x^2 + b*x^3 + c*x^4]) - ((5*
b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(2*a^2*(b^2 - 4*a*c)*x^3) + (b*(15*b^
2 - 52*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*a^3*(b^2 - 4*a*c)*x^2) - (3*(5*b^2 -
4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(8*a^(
7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.484992, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*x*Sqrt[a*x^2 + b*x^3 + c*x^4]) - ((5*
b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(2*a^2*(b^2 - 4*a*c)*x^3) + (b*(15*b^
2 - 52*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*a^3*(b^2 - 4*a*c)*x^2) - (3*(5*b^2 -
4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(8*a^(
7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 79.8979, size = 221, normalized size = 1.06 \[ \frac{2 \left (- 2 a c + b^{2} + b c x\right )}{a x \left (- 4 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}} - \frac{\left (- 12 a c + 5 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{2 a^{2} x^{3} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 52 a c + 15 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 a^{3} x^{2} \left (- 4 a c + b^{2}\right )} - \frac{3 x \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8 a^{\frac{7}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*a*c + b**2 + b*c*x)/(a*x*(-4*a*c + b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)) -
(-12*a*c + 5*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(2*a**2*x**3*(-4*a*c + b**2))
+ b*(-52*a*c + 15*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(4*a**3*x**2*(-4*a*c + b
**2)) - 3*x*(-4*a*c + 5*b**2)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a
)*sqrt(a + b*x + c*x**2)))/(8*a**(7/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.569977, size = 220, normalized size = 1.05 \[ -\frac{-3 x^2 \log (x) \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)}+3 x^2 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} \left (8 a^3 c-2 a^2 \left (b^2+10 b c x-12 c^2 x^2\right )+a b x \left (5 b^2-62 b c x-52 c^2 x^2\right )+15 b^3 x^2 (b+c x)\right )}{8 a^{7/2} x \left (b^2-4 a c\right ) \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

-(-2*Sqrt[a]*(8*a^3*c + 15*b^3*x^2*(b + c*x) + a*b*x*(5*b^2 - 62*b*c*x - 52*c^2*
x^2) - 2*a^2*(b^2 + 10*b*c*x - 12*c^2*x^2)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2
)*x^2*Sqrt[a + x*(b + c*x)]*Log[x] + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^2*Sqr
t[a + x*(b + c*x)]*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(8*a^(7/2)*
(b^2 - 4*a*c)*x*Sqrt[x^2*(a + x*(b + c*x))])

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 292, normalized size = 1.4 \[ -{\frac{x \left ( c{x}^{2}+bx+a \right ) }{32\,ac-8\,{b}^{2}} \left ( 48\,{a}^{7/2}{x}^{2}{c}^{2}-104\,{a}^{5/2}{x}^{3}b{c}^{2}+16\,{a}^{9/2}c-40\,{a}^{7/2}xbc-124\,{a}^{5/2}{x}^{2}{b}^{2}c+30\,{a}^{3/2}{x}^{3}{b}^{3}c-4\,{a}^{7/2}{b}^{2}+10\,{a}^{5/2}x{b}^{3}+30\,{a}^{3/2}{x}^{2}{b}^{4}-48\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}{a}^{3}{c}^{2}+72\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}{a}^{2}{b}^{2}c-15\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}a{b}^{4} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*x*(c*x^2+b*x+a)*(48*a^(7/2)*x^2*c^2-104*a^(5/2)*x^3*b*c^2+16*a^(9/2)*c-40*a
^(7/2)*x*b*c-124*a^(5/2)*x^2*b^2*c+30*a^(3/2)*x^3*b^3*c-4*a^(7/2)*b^2+10*a^(5/2)
*x*b^3+30*a^(3/2)*x^2*b^4-48*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(c*x^
2+b*x+a)^(1/2)*x^2*a^3*c^2+72*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(c*x
^2+b*x+a)^(1/2)*x^2*a^2*b^2*c-15*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*(
c*x^2+b*x+a)^(1/2)*x^2*a*b^4)/(c*x^4+b*x^3+a*x^2)^(3/2)/a^(9/2)/(4*a*c-b^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.353024, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt{a} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} +{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) + 4 \,{\left (2 \, a^{3} b^{2} - 8 \, a^{4} c -{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} -{\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{16 \,{\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} +{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}, \frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt{-a} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) - 2 \,{\left (2 \, a^{3} b^{2} - 8 \, a^{4} c -{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} -{\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{8 \,{\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} +{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^5 + (5*b^5 - 24*a*b^3*c + 16*
a^2*b*c^2)*x^4 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^3)*sqrt(a)*log(-(4*sqrt
(c*x^4 + b*x^3 + a*x^2)*(a*b*x + 2*a^2) + (8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2
*x)*sqrt(a))/x^3) + 4*(2*a^3*b^2 - 8*a^4*c - (15*a*b^3*c - 52*a^2*b*c^2)*x^3 - (
15*a*b^4 - 62*a^2*b^2*c + 24*a^3*c^2)*x^2 - 5*(a^2*b^3 - 4*a^3*b*c)*x)*sqrt(c*x^
4 + b*x^3 + a*x^2))/((a^4*b^2*c - 4*a^5*c^2)*x^5 + (a^4*b^3 - 4*a^5*b*c)*x^4 + (
a^5*b^2 - 4*a^6*c)*x^3), 1/8*(3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^5 + (5*
b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^4 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^3
)*sqrt(-a)*arctan(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x^4 + b*x^3 + a*x^2)*a))
- 2*(2*a^3*b^2 - 8*a^4*c - (15*a*b^3*c - 52*a^2*b*c^2)*x^3 - (15*a*b^4 - 62*a^2*
b^2*c + 24*a^3*c^2)*x^2 - 5*(a^2*b^3 - 4*a^3*b*c)*x)*sqrt(c*x^4 + b*x^3 + a*x^2)
)/((a^4*b^2*c - 4*a^5*c^2)*x^5 + (a^4*b^3 - 4*a^5*b*c)*x^4 + (a^5*b^2 - 4*a^6*c)
*x^3)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.614045, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x