3.357 \(\int \frac{x^4}{a-b x^3} \, dx\)

Optimal. Leaf size=125 \[ \frac{a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{x^2}{2 b} \]

[Out]

-x^2/(2*b) - (a^(2/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3
]*b^(5/3)) - (a^(2/3)*Log[a^(1/3) - b^(1/3)*x])/(3*b^(5/3)) + (a^(2/3)*Log[a^(2/
3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(5/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.153958, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a - b*x^3),x]

[Out]

-x^2/(2*b) - (a^(2/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3
]*b^(5/3)) - (a^(2/3)*Log[a^(1/3) - b^(1/3)*x])/(3*b^(5/3)) + (a^(2/3)*Log[a^(2/
3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(5/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.4006, size = 116, normalized size = 0.93 \[ - \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}}} + \frac{a^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}}} - \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}}} - \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-b*x**3+a),x)

[Out]

-a**(2/3)*log(a**(1/3) - b**(1/3)*x)/(3*b**(5/3)) + a**(2/3)*log(a**(2/3) + a**(
1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(5/3)) - sqrt(3)*a**(2/3)*atan(sqrt(3)*(a
**(1/3)/3 + 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(5/3)) - x**2/(2*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0698926, size = 111, normalized size = 0.89 \[ -\frac{-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+3 b^{2/3} x^2}{6 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a - b*x^3),x]

[Out]

-(3*b^(2/3)*x^2 + 2*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]]
+ 2*a^(2/3)*Log[a^(1/3) - b^(1/3)*x] - a^(2/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(6*b^(5/3))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 103, normalized size = 0.8 \[ -{\frac{{x}^{2}}{2\,b}}-{\frac{a}{3\,{b}^{2}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a}{6\,{b}^{2}}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-b*x^3+a),x)

[Out]

-1/2*x^2/b-1/3*a/b^2/(a/b)^(1/3)*ln(x-(a/b)^(1/3))+1/6*a/b^2/(a/b)^(1/3)*ln(x^2+
x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x+1))

_______________________________________________________________________________________

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^4/(b*x^3 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233852, size = 200, normalized size = 1.6 \[ -\frac{\sqrt{3}{\left (3 \, \sqrt{3} x^{2} + \sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} + b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x - b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 6 \, \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x + \sqrt{3} b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(3*sqrt(3)*x^2 + sqrt(3)*(-a^2/b^2)^(1/3)*log(a*x^2 + b*x*(-a^2/b^
2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 2*sqrt(3)*(-a^2/b^2)^(1/3)*log(a*x - b*(-a^2/b^
2)^(2/3)) - 6*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x + sqrt(3)*b*(-a^2/b^2)^
(2/3))/(b*(-a^2/b^2)^(2/3))))/b

_______________________________________________________________________________________

Sympy [A]  time = 1.25902, size = 34, normalized size = 0.27 \[ - \operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (- \frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-b*x**3+a),x)

[Out]

-RootSum(27*_t**3*b**5 - a**2, Lambda(_t, _t*log(-9*_t**2*b**3/a + x))) - x**2/(
2*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.236371, size = 143, normalized size = 1.14 \[ -\frac{x^{2}}{2 \, b} - \frac{\left (\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} - \frac{\sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{3}} + \frac{\left (a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^2/b - 1/3*(a/b)^(2/3)*ln(abs(x - (a/b)^(1/3)))/b - 1/3*sqrt(3)*(a*b^2)^(2
/3)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/b^3 + 1/6*(a*b^2)^(2/3)*
ln(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/b^3