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))

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Rubi [A]  time = 0.066276, 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, Rules used = {321, 292, 31, 634, 617, 204, 628} \[ \frac{a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}-\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{x^2}{2 b} \]

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))

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{a-b x^3} \, dx &=-\frac{x^2}{2 b}+\frac{a \int \frac{x}{a-b x^3} \, dx}{b}\\ &=-\frac{x^2}{2 b}+\frac{a^{2/3} \int \frac{1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 b^{4/3}}-\frac{a^{2/3} \int \frac{\sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^{4/3}}\\ &=-\frac{x^2}{2 b}-\frac{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \int \frac{\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{5/3}}-\frac{a \int \frac{1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{4/3}}\\ &=-\frac{x^2}{2 b}-\frac{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}+\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} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{5/3}}\\ &=-\frac{x^2}{2 b}-\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{a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{5/3}}+\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}}\\ \end{align*}

Mathematica [A]  time = 0.0334551, 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))

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Maple [A]  time = 0.003, size = 103, normalized size = 0.8 \begin{align*} -{\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}+\sqrt [3]{{\frac{a}{b}}}x+ \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}}}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 1.53103, size = 309, normalized size = 2.47 \begin{align*} -\frac{3 \, x^{2} + 2 \, \sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + \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 \, \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 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.375987, size = 34, normalized size = 0.27 \begin{align*} - \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} \end{align*}

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)

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Giac [A]  time = 1.4327, size = 143, normalized size = 1.14 \begin{align*} -\frac{x^{2}}{2 \, b} - \frac{\left (\frac{a}{b}\right )^{\frac{2}{3}} \log \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}} \log \left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{3}} \end{align*}

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)*log(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)*log(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/b^3