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

Optimal. Leaf size=70 \[ \frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac {2 C \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{-b}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1866, 31, 617, 204} \[ \frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac {2 C \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1866

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] && PolyQ[P2, x,
2]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 116, normalized size = 1.66 \[ -\frac {C \left (-b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+(-b)^{2/3} \log \left (a+b x^3\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.89, size = 205, normalized size = 2.93 \[ \left [\frac {\sqrt {\frac {1}{3}} C b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, b x^{3} + 3 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {1}{3}} b x^{2} + a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x + a \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} - a}{b x^{3} + a}\right ) - C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}, -\frac {2 \, \sqrt {\frac {1}{3}} C b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x + a \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.05, size = 122, normalized size = 1.74 \[ -\frac {2 \sqrt {3}\, C \,a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {2 C \,a^{\frac {2}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {C \,a^{\frac {2}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\left (-b \right )^{\frac {2}{3}} C \ln \left (b \,x^{3}+a \right )}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 3.02, size = 173, normalized size = 2.47 \[ \frac {2 \, \sqrt {3} {\left (C a \left (-b\right )^{\frac {2}{3}} - {\left (3 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {C a \left (-b\right )^{\frac {2}{3}}}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} - \frac {{\left (C \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - C a^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (C \left (-b\right )^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C a^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*sqrt(3)*(C*a*(-b)^(2/3) - (3*C*a^(2/3)*(a/b)^(1/3) + C*a*(-b)^(2/3)/b)*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^
(1/3))/(a/b)^(1/3))/(a*b) - 1/3*(C*(-b)^(2/3)*(a/b)^(2/3) - C*a^(2/3))*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/
(b*(a/b)^(2/3)) - 1/3*(C*(-b)^(2/3)*(a/b)^(2/3) + 2*C*a^(2/3))*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))

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mupad [B]  time = 5.24, size = 221, normalized size = 3.16 \[ \sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right )\,\left (\frac {6\,C\,a}{{\left (-b\right )}^{4/3}}+\frac {\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right )\,a\,9}{b}-\frac {6\,C\,a^{2/3}\,x}{b}\right )-\frac {C^2\,a}{{\left (-b\right )}^{5/3}}-\frac {2\,C^2\,a^{2/3}\,x}{{\left (-b\right )}^{4/3}}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z+9\,C^3\,a^2\,b^2,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(root(27*a^2*b^3*z^3 + 27*C*a^2*(-b)^(8/3)*z^2 - 9*C^2*a^2*(-b)^(7/3)*z + 9*C^3*a^2*b^2, z, k)*((6*C
*a)/(-b)^(4/3) + (9*root(27*a^2*b^3*z^3 + 27*C*a^2*(-b)^(8/3)*z^2 - 9*C^2*a^2*(-b)^(7/3)*z + 9*C^3*a^2*b^2, z,
k)*a)/b - (6*C*a^(2/3)*x)/b) - (C^2*a)/(-b)^(5/3) - (2*C^2*a^(2/3)*x)/(-b)^(4/3))*root(27*a^2*b^3*z^3 + 27*C*
a^2*(-b)^(8/3)*z^2 - 9*C^2*a^2*(-b)^(7/3)*z + 9*C^3*a^2*b^2, z, k), k, 1, 3)

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sympy [A]  time = 1.24, size = 73, normalized size = 1.04 \[ - \operatorname {RootSum} {\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac {2}{3}} + t C^{2} \left (- b\right )^{\frac {4}{3}} - C^{3} b, \left (t \mapsto t \log {\left (\frac {3 t \sqrt [3]{a}}{2 C} - \frac {\sqrt [3]{a} \left (- b\right )^{\frac {2}{3}}}{2 b} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(3*_t**3*b**2 - 3*_t**2*C*b*(-b)**(2/3) + _t*C**2*(-b)**(4/3) - C**3*b, Lambda(_t, _t*log(3*_t*a**(1/3
)/(2*C) - a**(1/3)*(-b)**(2/3)/(2*b) + x)))

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