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]

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

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Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 31

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

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])

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 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]

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*}

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Mathematica [B]  time = 0.04, size = 106, normalized size = 2.26 \[ \frac {C \left (-a^{2/3} \log \left (8 x^3-a\right )+(-a)^{2/3} \log \left (a^{2/3}+2 \sqrt [3]{a} x+4 x^2\right )-2 (-a)^{2/3} \log \left (\sqrt [3]{a}-2 x\right )+2 \sqrt {3} (-a)^{2/3} \tan ^{-1}\left (\frac {\frac {4 x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )}{12 a^{2/3}} \]

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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fricas [A]  time = 0.55, size = 43, normalized size = 0.91 \[ \frac {1}{6} \, \sqrt {3} C \arctan \left (\frac {4 \, \sqrt {3} \left (-a\right )^{\frac {2}{3}} x + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{4} \, C \log \left (2 \, x + \left (-a\right )^{\frac {1}{3}}\right ) \]

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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giac [B]  time = 0.21, size = 98, normalized size = 2.09 \[ \frac {\sqrt {3} {\left (\sqrt {3} i {\left | a \right |} - a\right )} C \arctan \left (\frac {\sqrt {3} {\left (4 \, x + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{12 \, a} + \frac {{\left (\sqrt {3} i {\left | a \right |} - 3 \, a\right )} C \log \left (x^{2} + \frac {1}{2} \, a^{\frac {1}{3}} x + \frac {1}{4} \, a^{\frac {2}{3}}\right )}{24 \, a} - \frac {{\left (2 \, C \left (-a\right )^{\frac {2}{3}} + C a^{\frac {2}{3}}\right )} \log \left ({\left | x - \frac {1}{2} \, a^{\frac {1}{3}} \right |}\right )}{12 \, a^{\frac {2}{3}}} \]

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)

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maple [B]  time = 0.05, size = 110, normalized size = 2.34 \[ -\frac {C \ln \left (8 x^{3}-a \right )}{12}+\frac {\left (-a \right )^{\frac {2}{3}} 8^{\frac {2}{3}} \sqrt {3}\, C \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,8^{\frac {1}{3}} x}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{24 a^{\frac {2}{3}}}-\frac {\left (-a \right )^{\frac {2}{3}} 8^{\frac {2}{3}} C \ln \left (x -\frac {8^{\frac {2}{3}} a^{\frac {1}{3}}}{8}\right )}{24 a^{\frac {2}{3}}}+\frac {\left (-a \right )^{\frac {2}{3}} 8^{\frac {2}{3}} C \ln \left (x^{2}+\frac {8^{\frac {2}{3}} a^{\frac {1}{3}} x}{8}+\frac {8^{\frac {1}{3}} a^{\frac {2}{3}}}{8}\right )}{48 a^{\frac {2}{3}}} \]

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 [B]  time = 2.99, size = 93, normalized size = 1.98 \[ \frac {\sqrt {3} C \left (-a\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (4 \, x + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{6 \, a^{\frac {2}{3}}} + \frac {{\left (C \left (-a\right )^{\frac {2}{3}} - C a^{\frac {2}{3}}\right )} \log \left (4 \, x^{2} + 2 \, a^{\frac {1}{3}} x + a^{\frac {2}{3}}\right )}{12 \, a^{\frac {2}{3}}} - \frac {{\left (2 \, C \left (-a\right )^{\frac {2}{3}} + C a^{\frac {2}{3}}\right )} \log \left (x - \frac {1}{2} \, a^{\frac {1}{3}}\right )}{12 \, a^{\frac {2}{3}}} \]

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]

1/6*sqrt(3)*C*(-a)^(2/3)*arctan(1/3*sqrt(3)*(4*x + a^(1/3))/a^(1/3))/a^(2/3) + 1/12*(C*(-a)^(2/3) - C*a^(2/3))
*log(4*x^2 + 2*a^(1/3)*x + a^(2/3))/a^(2/3) - 1/12*(2*C*(-a)^(2/3) + C*a^(2/3))*log(x - 1/2*a^(1/3))/a^(2/3)

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mupad [B]  time = 0.33, size = 142, normalized size = 3.02 \[ \sum _{k=1}^3\ln \left (-\frac {\left (C+12\,\mathrm {root}\left (1728\,a^2\,z^3+432\,C\,a^2\,z^2+36\,C^2\,a^2\,z+9\,C^3\,a^2,z,k\right )\right )\,\left (C\,a+\mathrm {root}\left (1728\,a^2\,z^3+432\,C\,a^2\,z^2+36\,C^2\,a^2\,z+9\,C^3\,a^2,z,k\right )\,a\,12+4\,C\,{\left (-a\right )}^{2/3}\,x\right )}{128}\right )\,\mathrm {root}\left (1728\,a^2\,z^3+432\,C\,a^2\,z^2+36\,C^2\,a^2\,z+9\,C^3\,a^2,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-((C + 12*root(1728*a^2*z^3 + 432*C*a^2*z^2 + 36*C^2*a^2*z + 9*C^3*a^2, z, k))*(C*a + 12*root(1728*
a^2*z^3 + 432*C*a^2*z^2 + 36*C^2*a^2*z + 9*C^3*a^2, z, k)*a + 4*C*(-a)^(2/3)*x))/128)*root(1728*a^2*z^3 + 432*
C*a^2*z^2 + 36*C^2*a^2*z + 9*C^3*a^2, z, k), k, 1, 3)

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sympy [C]  time = 0.93, size = 95, normalized size = 2.02 \[ - C \left (\frac {\log {\left (- \frac {a}{2 \left (- a\right )^{\frac {2}{3}}} + x \right )}}{4} + \frac {\sqrt {3} i \log {\left (\frac {a}{4 \left (- a\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{4 \left (- a\right )^{\frac {2}{3}}} + x \right )}}{12} - \frac {\sqrt {3} i \log {\left (\frac {a}{4 \left (- a\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{4 \left (- a\right )^{\frac {2}{3}}} + x \right )}}{12}\right ) \]

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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