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

Optimal. Leaf size=48 \[ \frac{C \log \left (\sqrt [3]{b} x+2\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

[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 = 0.0369016, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1863, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{b} x+2\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

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 1863

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{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 = 0.018627, size = 76, normalized size = 1.58 \[ \frac{C \left (-\log \left (b^{2/3} x^2-2 \sqrt [3]{b} x+4\right )+\log \left (b x^3+8\right )+2 \log \left (\sqrt [3]{b} x+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b} x-1}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.044, size = 117, normalized size = 2.4 \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}{\frac{1}{\sqrt [3]{b}}}} \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^(1/3)*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 = 1.08095, size = 387, normalized size = 8.06 \begin{align*} \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{b x^{3} + 6 \, \sqrt{\frac{1}{3}}{\left (b x^{2} + b^{\frac{2}{3}} x - 2 \, b^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - 6 \, b^{\frac{1}{3}} x - 4}{b x^{3} + 8}\right ) + C b^{\frac{2}{3}} \log \left (b x + 2 \, b^{\frac{2}{3}}\right )}{b}, \frac{2 \, \sqrt{\frac{1}{3}} C b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (b^{\frac{2}{3}} x - b^{\frac{1}{3}}\right )}}{b^{\frac{1}{3}}}\right ) + C b^{\frac{2}{3}} \log \left (b x + 2 \, b^{\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(-1/b^(2/3))*log((b*x^3 + 6*sqrt(1/3)*(b*x^2 + b^(2/3)*x - 2*b^(1/3))*sqrt(-1/b^(2/3)) - 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^(2/3)*arctan(sqrt(1/3)*(b^(
2/3)*x - b^(1/3))/b^(1/3)) + C*b^(2/3)*log(b*x + 2*b^(2/3)))/b]

________________________________________________________________________________________

Sympy [A]  time = 0.49057, size = 58, normalized size = 1.21 \begin{align*} \operatorname{RootSum}{\left (3 t^{3} b^{\frac{5}{3}} - 3 t^{2} C b^{\frac{4}{3}} + t C^{2} b - C^{3} b^{\frac{2}{3}}, \left ( t \mapsto t \log{\left (x + \frac{3 t \sqrt [3]{b} - C}{C \sqrt [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**(5/3) - 3*_t**2*C*b**(4/3) + _t*C**2*b - C**3*b**(2/3), Lambda(_t, _t*log(x + (3*_t*b**(1/3
) - C)/(C*b**(1/3)))))

________________________________________________________________________________________

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