3.4.34 \(\int \frac {x^4}{(a+b x^3)^2} \, dx\) [334]

Optimal. Leaf size=136 \[ -\frac {x^2}{3 b \left (a+b x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}} \]

[Out]

-1/3*x^2/b/(b*x^3+a)-2/9*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(5/3)+1/9*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a
^(1/3)/b^(5/3)-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b^(5/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {294, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {2 \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac {x^2}{3 b \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^3)^2,x]

[Out]

-1/3*x^2/(b*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(1/3)*b^(5/3)) -
(2*Log[a^(1/3) + b^(1/3)*x])/(9*a^(1/3)*b^(5/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(9*a^(1/3)*
b^(5/3))

Rule 31

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

Rule 210

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

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

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 631

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 642

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]

Rule 648

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]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b x^3\right )^2} \, dx &=-\frac {x^2}{3 b \left (a+b x^3\right )}+\frac {2 \int \frac {x}{a+b x^3} \, dx}{3 b}\\ &=-\frac {x^2}{3 b \left (a+b x^3\right )}-\frac {2 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}+\frac {2 \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}{9 \sqrt [3]{a} b^{4/3}}\\ &=-\frac {x^2}{3 b \left (a+b x^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {\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}{9 \sqrt [3]{a} b^{5/3}}+\frac {\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^{4/3}}\\ &=-\frac {x^2}{3 b \left (a+b x^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{5/3}}\\ &=-\frac {x^2}{3 b \left (a+b x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 119, normalized size = 0.88 \begin {gather*} \frac {-\frac {3 b^{2/3} x^2}{a+b x^3}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^3)^2,x]

[Out]

((-3*b^(2/3)*x^2)/(a + b*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) - (2*Log[a^(1/
3) + b^(1/3)*x])/a^(1/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/a^(1/3))/(9*b^(5/3))

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 114, normalized size = 0.84

method	result	size
risch	\(-\frac {x^{2}}{3 b \left (b \,x^{3}+a \right )}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{9 b^{2}}\)	\(45\)
default	\(-\frac {x^{2}}{3 b \left (b \,x^{3}+a \right )}+\frac {-\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 116, normalized size = 0.85 \begin {gather*} -\frac {x^{2}}{3 \, {\left (b^{2} x^{3} + a b\right )}} + \frac {2 \, \sqrt {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 )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/3*x^2/(b^2*x^3 + a*b) + 2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(1/3)) +
1/9*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(1/3)) - 2/9*log(x + (a/b)^(1/3))/(b^2*(a/b)^(1/3))

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 400, normalized size = 2.94 \begin {gather*} \left [-\frac {3 \, a b^{2} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}, -\frac {3 \, a b^{2} x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[-1/9*(3*a*b^2*x^2 - 3*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)
*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) -
(b*x^3 + a)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 2*(b*x^3 + a)*(-a*b^2)^(2/3)*
log(b*x - (-a*b^2)^(1/3)))/(a*b^4*x^3 + a^2*b^3), -1/9*(3*a*b^2*x^2 - 6*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*sqrt(-(-
a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) - (b*x^3 + a)*(-a*b^2)^(2
/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 2*(b*x^3 + a)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)
))/(a*b^4*x^3 + a^2*b^3)]

________________________________________________________________________________________

Sympy [A]
time = 0.11, size = 44, normalized size = 0.32 \begin {gather*} - \frac {x^{2}}{3 a b + 3 b^{2} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a b^{3}}{4} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2/(3*a*b + 3*b**2*x**3) + RootSum(729*_t**3*a*b**5 + 8, Lambda(_t, _t*log(81*_t**2*a*b**3/4 + x)))

________________________________________________________________________________________

Giac [A]
time = 1.05, size = 132, normalized size = 0.97 \begin {gather*} -\frac {x^{2}}{3 \, {\left (b x^{3} + a\right )} b} - \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b} - \frac {2 \, \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 )}{9 \, a 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 )}{9 \, a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/3*x^2/((b*x^3 + a)*b) - 2/9*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) - 2/9*sqrt(3)*(-a*b^2)^(2/3)*arct
an(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/9*(-a*b^2)^(2/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/
b)^(2/3))/(a*b^3)

________________________________________________________________________________________

Mupad [B]
time = 0.22, size = 138, normalized size = 1.01 \begin {gather*} \frac {2\,\ln \left (\frac {4\,x}{9\,b}-\frac {4\,{\left (-a\right )}^{1/3}}{9\,b^{4/3}}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {x^2}{3\,b\,\left (b\,x^3+a\right )}+\frac {\ln \left (\frac {4\,x}{9\,b}-\frac {{\left (-a\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,b^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {\ln \left (\frac {4\,x}{9\,b}-\frac {{\left (-a\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,b^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^3)^2,x)

[Out]

(2*log((4*x)/(9*b) - (4*(-a)^(1/3))/(9*b^(4/3))))/(9*(-a)^(1/3)*b^(5/3)) - x^2/(3*b*(a + b*x^3)) + (log((4*x)/
(9*b) - ((-a)^(1/3)*(3^(1/2)*1i - 1)^2)/(9*b^(4/3)))*(3^(1/2)*1i - 1))/(9*(-a)^(1/3)*b^(5/3)) - (log((4*x)/(9*
b) - ((-a)^(1/3)*(3^(1/2)*1i + 1)^2)/(9*b^(4/3)))*(3^(1/2)*1i + 1))/(9*(-a)^(1/3)*b^(5/3))

________________________________________________________________________________________