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]

(-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)

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Rubi [A]  time = 0.0718664, antiderivative size = 70, normalized size of antiderivative = 1., 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 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]

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{-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*}

Mathematica [A]  time = 0.0283125, 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]

-(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]))/(3*b)

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Maple [B]  time = 0.007, size = 122, normalized size = 1.7 \begin{align*} -{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08685, size = 539, normalized size = 7.7 \begin{align*} \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 ] \end{align*}

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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Sympy [A]  time = 0.765978, size = 73, normalized size = 1.04 \begin{align*} - \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 )} \end{align*}

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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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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