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

Optimal. Leaf size=57 \[ \frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{-b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{-b}}-\frac{C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0689679, antiderivative size = 57, 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} \[ \frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{-b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{-b}}-\frac{C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0291527, size = 99, normalized size = 1.74 \[ \frac{C \left (-b^{2/3} \log \left (b^{2/3} x^2+2 \sqrt [3]{b} x+4\right )+2 b^{2/3} \log \left (2-\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b} x+1}{\sqrt{3}}\right )+(-b)^{2/3} \log \left (8-b x^3\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(-2*Sqrt[3]*b^(2/3)*ArcTan[(1 + b^(1/3)*x)/Sqrt[3]] + 2*b^(2/3)*Log[2 - b^(1/3)*x] - b^(2/3)*Log[4 + 2*b^(1
/3)*x + b^(2/3)*x^2] + (-b)^(2/3)*Log[8 - b*x^3]))/(3*b)

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 122, normalized size = 2.1 \begin{align*}{\frac{C\sqrt [3]{8}}{3\,b}\ln \left ( x-\sqrt [3]{8}\sqrt [3]{{b}^{-1}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\sqrt [3]{8}}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{8}\sqrt [3]{{b}^{-1}}x+{8}^{{\frac{2}{3}}} \left ({b}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\sqrt [3]{8}\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{8}^{{\frac{2}{3}}}x}{4}{\frac{1}{\sqrt [3]{{b}^{-1}}}}}+1 \right ) } \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}-8 \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*C/b*8^(1/3)/(1/b)^(2/3)*ln(x-8^(1/3)*(1/b)^(1/3))-1/6*C/b*8^(1/3)/(1/b)^(2/3)*ln(x^2+8^(1/3)*(1/b)^(1/3)*x
+8^(2/3)*(1/b)^(2/3))-1/3*C/b*8^(1/3)/(1/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(1/4*8^(2/3)/(1/b)^(1/3)*x+1))+1/
3*C*(-b)^(2/3)/b*ln(b*x^3-8)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.05274, size = 466, normalized size = 8.18 \begin{align*} \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{b x^{3} - 6 \, \sqrt{\frac{1}{3}}{\left (b x^{2} - \left (-b\right )^{\frac{2}{3}} x + 2 \, \left (-b\right )^{\frac{1}{3}}\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 6 \, \left (-b\right )^{\frac{1}{3}} x + 4}{b x^{3} - 8}\right ) + C \left (-b\right )^{\frac{2}{3}} \log \left (b x - 2 \, \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 (\sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{2}{3}} x - \left (-b\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}\right ) - C \left (-b\right )^{\frac{2}{3}} \log \left (b x - 2 \, \left (-b\right )^{\frac{2}{3}}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(1/3)*C*b*sqrt((-b)^(1/3)/b)*log((b*x^3 - 6*sqrt(1/3)*(b*x^2 - (-b)^(2/3)*x + 2*(-b)^(1/3))*sqrt((-b)^(1
/3)/b) + 6*(-b)^(1/3)*x + 4)/(b*x^3 - 8)) + C*(-b)^(2/3)*log(b*x - 2*(-b)^(2/3)))/b, -(2*sqrt(1/3)*C*b*sqrt(-(
-b)^(1/3)/b)*arctan(sqrt(1/3)*((-b)^(2/3)*x - (-b)^(1/3))*sqrt(-(-b)^(1/3)/b)) - C*(-b)^(2/3)*log(b*x - 2*(-b)
^(2/3)))/b]

________________________________________________________________________________________

Sympy [A]  time = 0.582586, size = 58, normalized size = 1.02 \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}{C} + x + \frac{\left (- b\right )^{\frac{2}{3}}}{b} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError