### 3.144 $$\int \frac{\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx$$

Optimal. Leaf size=47 $-\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}}$

[Out]

(-3*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))

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Rubi [A]  time = 0.062168, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 59, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.051, Rules used = {1594, 1468, 628} $-\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}}$

Antiderivative was successfully veriﬁed.

[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 1594

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]

Rule 1468

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 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{\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 &=\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 \operatorname{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]  time = 0.0198506, size = 47, normalized size = 1. $-\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}}$

Antiderivative was successfully veriﬁed.

[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))

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -3\,{\frac{\ln \left ({c}^{2/3}{d}^{2/3}\sqrt [3]{x}-\sqrt [3]{c}{x}^{2/3}d-c\sqrt [3]{d} \right ) }{{d}^{2/3}\sqrt [3]{c}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]  time = 1.06346, size = 46, normalized size = 0.98 \begin{align*} -\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}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 1.38533, size = 109, normalized size = 2.32 \begin{align*} -\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}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [C]  time = 7.14819, size = 126, normalized size = 2.68 \begin{align*} - \frac{3 \log{\left (- \frac{\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} - \frac{\sqrt{3} i \sqrt{c^{\frac{4}{3}}} \sqrt{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} i \sqrt{c^{\frac{4}{3}}} \sqrt{d^{\frac{4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac{2}{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)*I*sqrt(c**(4/3))*sqrt(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)*I*sqrt(c**(4/3))*sqrt(d**(4/3))/(2*c**(1/3)*d))/(
c**(1/3)*d**(2/3))

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Giac [A]  time = 1.16123, size = 46, normalized size = 0.98 \begin{align*} -\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}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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]

-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))