\(\int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx\) [144]

Optimal result

Integrand size = 59, antiderivative size = 47 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]

[Out]

-3*ln(c^(2/3)-c^(1/3)*d^(1/3)*x^(1/3)+d^(2/3)*x^(2/3))/c^(1/3)/d^(2/3)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1608, 1482, 642} \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]

[In]

Int[(c^(1/3) - 2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3) - c^(2/3)*d^(2/3)*x + c^(1/3)*d*x^(4/3)),x]

[Out]

(-3*Log[c^(2/3) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3)*x^(2/3)])/(c^(1/3)*d^(2/3))

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 1482

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]

[In]

Integrate[(c^(1/3) - 2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3) - c^(2/3)*d^(2/3)*x + c^(1/3)*d*x^(4/3)),x]

[Out]

(-3*Log[c^(2/3) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3)*x^(2/3)])/(c^(1/3)*d^(2/3))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {3 \ln \left (c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}}-c^{\frac {1}{3}} d \,x^{\frac {2}{3}}-c \,d^{\frac {1}{3}}\right )}{d^{\frac {2}{3}} c^{\frac {1}{3}}}\)	\(36\)
default	\(-\frac {3 \ln \left (c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}}-c^{\frac {1}{3}} d \,x^{\frac {2}{3}}-c \,d^{\frac {1}{3}}\right )}{d^{\frac {2}{3}} c^{\frac {1}{3}}}\)	\(36\)

[In]

int((c^(1/3)-2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3)-c^(2/3)*d^(2/3)*x+c^(1/3)*d*x^(4/3)),x,method=_RETURNVERBOS
E)

[Out]

-3/d^(2/3)/c^(1/3)*ln(c^(2/3)*d^(2/3)*x^(1/3)-c^(1/3)*d*x^(2/3)-c*d^(1/3))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \, \log \left (d x^{\frac {2}{3}} - c^{\frac {1}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}} + c^{\frac {2}{3}} d^{\frac {1}{3}}\right )}{c^{\frac {1}{3}} d^{\frac {2}{3}}} \]

[In]

integrate((c^(1/3)-2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3)-c^(2/3)*d^(2/3)*x+c^(1/3)*d*x^(4/3)),x, algorithm="fr
icas")

[Out]

-3*log(d*x^(2/3) - c^(1/3)*d^(2/3)*x^(1/3) + c^(2/3)*d^(1/3))/(c^(1/3)*d^(2/3))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (44) = 88\).

Time = 2.61 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=- \frac {3 \log {\left (- \frac {\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} - \frac {\sqrt {3} \sqrt {- c^{\frac {4}{3}} d^{\frac {4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac {2}{3}}} - \frac {3 \log {\left (- \frac {\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} + \frac {\sqrt {3} \sqrt {- c^{\frac {4}{3}} d^{\frac {4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac {2}{3}}} \]

[In]

integrate((c**(1/3)-2*d**(1/3)*x**(1/3))/(c*d**(1/3)*x**(2/3)-c**(2/3)*d**(2/3)*x+c**(1/3)*d*x**(4/3)),x)

[Out]

-3*log(-c**(1/3)/(2*d**(1/3)) + x**(1/3) - sqrt(3)*sqrt(-c**(4/3)*d**(4/3))/(2*c**(1/3)*d))/(c**(1/3)*d**(2/3)
) - 3*log(-c**(1/3)/(2*d**(1/3)) + x**(1/3) + sqrt(3)*sqrt(-c**(4/3)*d**(4/3))/(2*c**(1/3)*d))/(c**(1/3)*d**(2
/3))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \, \log \left (c^{\frac {1}{3}} d x^{\frac {2}{3}} - c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}} + c d^{\frac {1}{3}}\right )}{c^{\frac {1}{3}} d^{\frac {2}{3}}} \]

[In]

integrate((c^(1/3)-2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3)-c^(2/3)*d^(2/3)*x+c^(1/3)*d*x^(4/3)),x, algorithm="ma
xima")

[Out]

-3*log(c^(1/3)*d*x^(2/3) - c^(2/3)*d^(2/3)*x^(1/3) + c*d^(1/3))/(c^(1/3)*d^(2/3))

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^(1/3)-2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3)-c^(2/3)*d^(2/3)*x+c^(1/3)*d*x^(4/3)),x, algorithm="gi
ac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{[%%%{%%{[%%%{1,[1]%%%},0]:[1,0,0,%%%{-1,[1]%%%}]%%},[
1]%%%},0]:[

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3\,\ln \left (x^{2/3}+\frac {c^{2/3}}{d^{2/3}}-\frac {c^{1/3}\,x^{1/3}}{d^{1/3}}\right )}{c^{1/3}\,d^{2/3}} \]

[In]

int((c^(1/3) - 2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3) - c^(2/3)*d^(2/3)*x + c^(1/3)*d*x^(4/3)),x)

[Out]

-(3*log(x^(2/3) + c^(2/3)/d^(2/3) - (c^(1/3)*x^(1/3))/d^(1/3)))/(c^(1/3)*d^(2/3))