\(\int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx\) [2841]

Optimal result

Integrand size = 26, antiderivative size = 290 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}-\frac {b c \arctan \left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{a}}}{x}\right )}{3 \sqrt {3} a^{4/3}}+\frac {d \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{9 a^{4/3}}-\frac {d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{18 a^{4/3}} \]

[Out]

1/3*c*x*(a*x^3-b)^(2/3)/a-1/9*b*c*arctan((1/3*x*3^(1/2)+2/3*(a*x^3-b)^(1/3)*3^(1/2)/a^(1/3))/x)*3^(1/2)/a^(4/3
)-1/3*d*arctan(-1/3*3^(1/2)+2/3*(a*x^3-b)^(1/3)*3^(1/2)/b^(1/3))*3^(1/2)/b^(1/3)+1/3*d*ln(b^(1/3)+(a*x^3-b)^(1
/3))/b^(1/3)-1/9*b*c*ln(-a^(1/3)*x+(a*x^3-b)^(1/3))/a^(4/3)-1/6*d*ln(b^(2/3)-b^(1/3)*(a*x^3-b)^(1/3)+(a*x^3-b)
^(2/3))/b^(1/3)+1/18*b*c*ln(a^(2/3)*x^2+a^(1/3)*x*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/a^(4/3)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.66, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1858, 272, 58, 631, 210, 31, 327, 245} \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {b c \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {b c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{6 a^{4/3}}+\frac {d \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c x \left (a x^3-b\right )^{2/3}}{3 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}} \]

[In]

Int[(-d + c*x^4)/(x*(-b + a*x^3)^(1/3)),x]

[Out]

(c*x*(-b + a*x^3)^(2/3))/(3*a) + (b*c*ArcTan[(1 + (2*a^(1/3)*x)/(-b + a*x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*a^(4/
3)) + (d*ArcTan[(b^(1/3) - 2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*b^(1/3)) - (d*Log[x])/(2*b^(1/3)
) + (d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(2*b^(1/3)) - (b*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/(6*a^(4/3
))

Rule 31

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

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

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 245

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 327

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 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 1858

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d}{x \sqrt [3]{-b+a x^3}}+\frac {c x^3}{\sqrt [3]{-b+a x^3}}\right ) \, dx \\ & = c \int \frac {x^3}{\sqrt [3]{-b+a x^3}} \, dx-d \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx \\ & = \frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {(b c) \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx}{3 a}-\frac {1}{3} d \text {Subst}\left (\int \frac {1}{x \sqrt [3]{-b+a x}} \, dx,x,x^3\right ) \\ & = \frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}}-\frac {1}{2} d \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{-b+a x^3}\right )+\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}} \\ & = \frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}}-\frac {d \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}} \\ & = \frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {1}{18} \left (\frac {6 c x \left (-b+a x^3\right )^{2/3}}{a}+\frac {6 \sqrt {3} d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {2 \sqrt {3} b c \arctan \left (\frac {1+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{a} x}}{\sqrt {3}}\right )}{a^{4/3}}+\frac {6 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{\sqrt [3]{b}}-\frac {2 b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{a^{4/3}}-\frac {3 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{\sqrt [3]{b}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{a^{4/3}}\right ) \]

[In]

Integrate[(-d + c*x^4)/(x*(-b + a*x^3)^(1/3)),x]

[Out]

((6*c*x*(-b + a*x^3)^(2/3))/a + (6*Sqrt[3]*d*ArcTan[(1 - (2*(-b + a*x^3)^(1/3))/b^(1/3))/Sqrt[3]])/b^(1/3) - (
2*Sqrt[3]*b*c*ArcTan[(1 + (2*(-b + a*x^3)^(1/3))/(a^(1/3)*x))/Sqrt[3]])/a^(4/3) + (6*d*Log[b^(1/3) + (-b + a*x
^3)^(1/3)])/b^(1/3) - (2*b*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/a^(4/3) - (3*d*Log[b^(2/3) - b^(1/3)*(-b
+ a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/b^(1/3) + (b*c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*
x^3)^(2/3)])/a^(4/3))/18

Maple [F]

\[\int \frac {c \,x^{4}-d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}d x\]

[In]

int((c*x^4-d)/x/(a*x^3-b)^(1/3),x)

[Out]

int((c*x^4-d)/x/(a*x^3-b)^(1/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\text {Timed out} \]

[In]

integrate((c*x^4-d)/x/(a*x^3-b)^(1/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.61 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.29 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=- \frac {c x^{4} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {7}{3}\right )} + \frac {d \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]

[In]

integrate((c*x**4-d)/x/(a*x**3-b)**(1/3),x)

[Out]

-c*x**4*exp(2*I*pi/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), a*x**3/b)/(3*b**(1/3)*gamma(7/3)) + d*gamma(1/3)*hy
per((1/3, 1/3), (4/3,), b*exp_polar(2*I*pi)/(a*x**3))/(3*a**(1/3)*x*gamma(4/3))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.86 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b \log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {4}{3}}} - \frac {6 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} b}{{\left (a^{2} - \frac {{\left (a x^{3} - b\right )} a}{x^{3}}\right )} x^{2}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \]

[In]

integrate((c*x^4-d)/x/(a*x^3-b)^(1/3),x, algorithm="maxima")

[Out]

-1/18*(2*sqrt(3)*b*arctan(1/3*sqrt(3)*(a^(1/3) + 2*(a*x^3 - b)^(1/3)/x)/a^(1/3))/a^(4/3) - b*log(a^(2/3) + (a*
x^3 - b)^(1/3)*a^(1/3)/x + (a*x^3 - b)^(2/3)/x^2)/a^(4/3) + 2*b*log(-a^(1/3) + (a*x^3 - b)^(1/3)/x)/a^(4/3) -
6*(a*x^3 - b)^(2/3)*b/((a^2 - (a*x^3 - b)*a/x^3)*x^2))*c - 1/6*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(a*x^3 - b)^(1
/3) - b^(1/3))/b^(1/3))/b^(1/3) + log((a*x^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/b^(1/3) - 2*log
((a*x^3 - b)^(1/3) + b^(1/3))/b^(1/3))*d

Giac [F]

\[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\int { \frac {c x^{4} - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x} \,d x } \]

[In]

integrate((c*x^4-d)/x/(a*x^3-b)^(1/3),x, algorithm="giac")

[Out]

integrate((c*x^4 - d)/((a*x^3 - b)^(1/3)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=-\int \frac {d-c\,x^4}{x\,{\left (a\,x^3-b\right )}^{1/3}} \,d x \]

[In]

int(-(d - c*x^4)/(x*(a*x^3 - b)^(1/3)),x)

[Out]

-int((d - c*x^4)/(x*(a*x^3 - b)^(1/3)), x)