3.49 \(\int \frac{x^3}{\sqrt{a x^2+b x^3+c x^4}} \, dx\)

Optimal. Leaf size=143 \[ \frac{x \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.291617, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 b \sqrt{a x^2+b x^3+c x^4}}{4 c^2 x}+\frac{\sqrt{a x^2+b x^3+c x^4}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.519, size = 129, normalized size = 0.9 \[ - \frac{3 b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 c^{2} x} + \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{2 c} + \frac{x \left (- a c + \frac{3 b^{2}}{4}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{5}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b*sqrt(a*x**2 + b*x**3 + c*x**4)/(4*c**2*x) + sqrt(a*x**2 + b*x**3 + c*x**4)/
(2*c) + x*(-a*c + 3*b**2/4)*sqrt(a + b*x + c*x**2)*atanh((b + 2*c*x)/(2*sqrt(c)*
sqrt(a + b*x + c*x**2)))/(2*c**(5/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.169847, size = 103, normalized size = 0.72 \[ \frac{x \left (\left (3 b^2-4 a c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{c} (2 c x-3 b) (a+x (b+c x))\right )}{8 c^{5/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*Sqrt[c]*(-3*b + 2*c*x)*(a + x*(b + c*x)) + (3*b^2 - 4*a*c)*Sqrt[a + x*(b +
c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(8*c^(5/2)*Sqrt[x^2*(a
+ x*(b + c*x))])

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 144, normalized size = 1. \[{\frac{x}{8}\sqrt{c{x}^{2}+bx+a} \left ( 4\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x-6\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}b-4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x*(c*x^2+b*x+a)^(1/2)*(4*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x-6*(c*x^2+b*x+a)^(1/2)
*c^(3/2)*b-4*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*c^2+3*ln(
1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^2*c)/(c*x^4+b*x^3+a*x^2)^
(1/2)/c^(7/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.299294, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt{c} x \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} -{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x - 3 \, b c\right )}}{16 \, c^{3} x}, -\frac{{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x - 3 \, b c\right )}}{8 \, c^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((3*b^2 - 4*a*c)*sqrt(c)*x*log((4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x +
b*c) - (8*c^2*x^3 + 8*b*c*x^2 + (b^2 + 4*a*c)*x)*sqrt(c))/x) - 4*sqrt(c*x^4 + b*
x^3 + a*x^2)*(2*c^2*x - 3*b*c))/(c^3*x), -1/8*((3*b^2 - 4*a*c)*sqrt(-c)*x*arctan
(1/2*(2*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*c)) - 2*sqrt(c*x^4 +
b*x^3 + a*x^2)*(2*c^2*x - 3*b*c))/(c^3*x)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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