3.32 \(\int \frac{(-a)^{2/3} C+2 C x^2}{a-8 x^3} \, dx\)

Optimal. Leaf size=47 \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]

[Out]

(C*ArcTan[(1 - (4*x)/(-a)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) - (C*Log[(-a)^(1/3) + 2*x])/4

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Rubi [A]  time = 0.061084, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1864, 31, 617, 204} \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]

Antiderivative was successfully verified.

[In]

Int[((-a)^(2/3)*C + 2*C*x^2)/(a - 8*x^3),x]

[Out]

(C*ArcTan[(1 - (4*x)/(-a)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) - (C*Log[(-a)^(1/3) + 2*x])/4

Rule 1864

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, With[{q = (-a)^(1/3)/(-b)^(1/3)}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x +
x^2), x], x]] /; EqQ[A*(-b)^(2/3) - (-a)^(1/3)*(-b)^(1/3)*B - 2*(-a)^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && Poly
Q[P2, x, 2]

Rule 31

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

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])

Rubi steps

\begin{align*} \int \frac{(-a)^{2/3} C+2 C x^2}{a-8 x^3} \, dx &=-\left (\frac{1}{4} C \int \frac{1}{\frac{\sqrt [3]{-a}}{2}+x} \, dx\right )-\frac{1}{8} \left (\sqrt [3]{-a} C\right ) \int \frac{1}{\frac{1}{4} (-a)^{2/3}-\frac{1}{2} \sqrt [3]{-a} x+x^2} \, dx\\ &=-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right )-\frac{1}{2} C \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{4 x}{\sqrt [3]{-a}}\right )\\ &=\frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right )\\ \end{align*}

Mathematica [B]  time = 0.0390344, size = 106, normalized size = 2.26 \[ \frac{C \left ((-a)^{2/3} \log \left (a^{2/3}+2 \sqrt [3]{a} x+4 x^2\right )-a^{2/3} \log \left (8 x^3-a\right )-2 (-a)^{2/3} \log \left (\sqrt [3]{a}-2 x\right )+2 \sqrt{3} (-a)^{2/3} \tan ^{-1}\left (\frac{\frac{4 x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )}{12 a^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[((-a)^(2/3)*C + 2*C*x^2)/(a - 8*x^3),x]

[Out]

(C*(2*Sqrt[3]*(-a)^(2/3)*ArcTan[(1 + (4*x)/a^(1/3))/Sqrt[3]] - 2*(-a)^(2/3)*Log[a^(1/3) - 2*x] + (-a)^(2/3)*Lo
g[a^(2/3) + 2*a^(1/3)*x + 4*x^2] - a^(2/3)*Log[-a + 8*x^3]))/(12*a^(2/3))

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Maple [B]  time = 0.007, size = 110, normalized size = 2.3 \begin{align*} -{\frac{C{8}^{{\frac{2}{3}}}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ( x-{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}}{48} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+{\frac{{8}^{{\frac{2}{3}}}x}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( 8\,{x}^{3}-a \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-a)^(2/3)*C+2*C*x^2)/(-8*x^3+a),x)

[Out]

-1/24*C*(-a)^(2/3)*8^(2/3)/a^(2/3)*ln(x-1/8*8^(2/3)*a^(1/3))+1/48*C*(-a)^(2/3)*8^(2/3)/a^(2/3)*ln(x^2+1/8*8^(2
/3)*a^(1/3)*x+1/8*8^(1/3)*a^(2/3))+1/24*C*(-a)^(2/3)*8^(2/3)/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*8^(1/3)/a^(
1/3)*x+1))-1/12*C*ln(8*x^3-a)

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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(((-a)^(2/3)*C+2*C*x^2)/(-8*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.02475, size = 130, normalized size = 2.77 \begin{align*} \frac{1}{6} \, \sqrt{3} C \arctan \left (\frac{4 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}} x + \sqrt{3} a}{3 \, a}\right ) - \frac{1}{4} \, C \log \left (2 \, x + \left (-a\right )^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-a)^(2/3)*C+2*C*x^2)/(-8*x^3+a),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*C*arctan(1/3*(4*sqrt(3)*(-a)^(2/3)*x + sqrt(3)*a)/a) - 1/4*C*log(2*x + (-a)^(1/3))

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Sympy [C]  time = 0.467872, size = 95, normalized size = 2.02 \begin{align*} - C \left (\frac{\log{\left (- \frac{a}{2 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{4} + \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12} - \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-a)**(2/3)*C+2*C*x**2)/(-8*x**3+a),x)

[Out]

-C*(log(-a/(2*(-a)**(2/3)) + x)/4 + sqrt(3)*I*log(a/(4*(-a)**(2/3)) - sqrt(3)*I*a/(4*(-a)**(2/3)) + x)/12 - sq
rt(3)*I*log(a/(4*(-a)**(2/3)) + sqrt(3)*I*a/(4*(-a)**(2/3)) + x)/12)

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Giac [B]  time = 1.13767, size = 132, normalized size = 2.81 \begin{align*} \frac{\sqrt{3}{\left (\sqrt{3} i{\left | a \right |} - a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{12 \, a} + \frac{{\left (\sqrt{3} i{\left | a \right |} - 3 \, a\right )} C \log \left (x^{2} + \frac{1}{2} \, a^{\frac{1}{3}} x + \frac{1}{4} \, a^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (2 \, C \left (-a\right )^{\frac{2}{3}} + C a^{\frac{2}{3}}\right )} \log \left ({\left | x - \frac{1}{2} \, a^{\frac{1}{3}} \right |}\right )}{12 \, a^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-a)^(2/3)*C+2*C*x^2)/(-8*x^3+a),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(sqrt(3)*i*abs(a) - a)*C*arctan(1/3*sqrt(3)*(4*x + a^(1/3))/a^(1/3))/a + 1/24*(sqrt(3)*i*abs(a) -
3*a)*C*log(x^2 + 1/2*a^(1/3)*x + 1/4*a^(2/3))/a - 1/12*(2*C*(-a)^(2/3) + C*a^(2/3))*log(abs(x - 1/2*a^(1/3)))
/a^(2/3)