3.372 \(\int \frac {x (2 \sqrt [3]{\frac {a}{b}} C+C x)}{a-b x^3} \, dx\)

Optimal. Leaf size=53 \[ -\frac {C \log \left (\sqrt [3]{\frac {a}{b}}-x\right )}{b}-\frac {2 C \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1869, 31, 617, 204} \[ -\frac {C \log \left (\sqrt [3]{\frac {a}{b}}-x\right )}{b}-\frac {2 C \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(2*(a/b)^(1/3)*C + C*x))/(a - b*x^3),x]

[Out]

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

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 1869

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/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 + (-(a/b))^(1/3)*B - 2*(-(a/b))^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.06, size = 147, normalized size = 2.77 \[ -\frac {C \left (-\sqrt [3]{b} \sqrt [3]{\frac {a}{b}} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\sqrt [3]{a} \log \left (a-b x^3\right )+2 \sqrt [3]{b} \sqrt [3]{\frac {a}{b}} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt {3} \sqrt [3]{b} \sqrt [3]{\frac {a}{b}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )\right )}{3 \sqrt [3]{a} b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(2*(a/b)^(1/3)*C + C*x))/(a - b*x^3),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 53, normalized size = 1.00 \[ -\frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 3 \, C \log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*(a/b)^(1/3)*C+C*x)/(-b*x^3+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.20, size = 90, normalized size = 1.70 \[ -\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {{\left (C b \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (a b^{2}\right )^{\frac {1}{3}} C \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*(a/b)^(1/3)*C+C*x)/(-b*x^3+a),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 90, normalized size = 1.70 \[ -\frac {2 \sqrt {3}\, C \arctan \left (\frac {\left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right ) \sqrt {3}}{3}\right )}{3 b}-\frac {2 C \ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b}+\frac {C \ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 b}-\frac {C \ln \left (b \,x^{3}-a \right )}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*(a/b)^(1/3)*C+C*x)/(-b*x^3+a),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.02, size = 52, normalized size = 0.98 \[ -\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {C \log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*(a/b)^(1/3)*C+C*x)/(-b*x^3+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.23, size = 155, normalized size = 2.92 \[ \sum _{k=1}^3\ln \left (-\frac {C^2\,a+{\mathrm {root}\left (27\,a\,b^3\,z^3+27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z+9\,C^3\,a,z,k\right )}^2\,a\,b^2\,9+C\,\mathrm {root}\left (27\,a\,b^3\,z^3+27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z+9\,C^3\,a,z,k\right )\,a\,b\,6-4\,C^2\,b\,x\,{\left (\frac {a}{b}\right )}^{2/3}}{b^3}\right )\,\mathrm {root}\left (27\,a\,b^3\,z^3+27\,C\,a\,b^2\,z^2+9\,C^2\,a\,b\,z+9\,C^3\,a,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(C*x + 2*C*(a/b)^(1/3)))/(a - b*x^3),x)

[Out]

symsum(log(-(C^2*a + 9*root(27*a*b^3*z^3 + 27*C*a*b^2*z^2 + 9*C^2*a*b*z + 9*C^3*a, z, k)^2*a*b^2 + 6*C*root(27
*a*b^3*z^3 + 27*C*a*b^2*z^2 + 9*C^2*a*b*z + 9*C^3*a, z, k)*a*b - 4*C^2*b*x*(a/b)^(2/3))/b^3)*root(27*a*b^3*z^3
+ 27*C*a*b^2*z^2 + 9*C^2*a*b*z + 9*C^3*a, z, k), k, 1, 3)

________________________________________________________________________________________

sympy [C]  time = 0.37, size = 102, normalized size = 1.92 \[ - \frac {C \left (\log {\left (- \frac {a}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} - \frac {\sqrt {3} i \log {\left (\frac {a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (\frac {a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*(a/b)**(1/3)*C+C*x)/(-b*x**3+a),x)

[Out]

-C*(log(-a/(b*(a/b)**(2/3)) + x) - sqrt(3)*I*log(a/(2*b*(a/b)**(2/3)) - sqrt(3)*I*a/(2*b*(a/b)**(2/3)) + x)/3
+ sqrt(3)*I*log(a/(2*b*(a/b)**(2/3)) + sqrt(3)*I*a/(2*b*(a/b)**(2/3)) + x)/3)/b

________________________________________________________________________________________