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

Optimal. Leaf size=47 \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0343279, antiderivative size = 47, 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} \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.023975, size = 72, normalized size = 1.53 \[ \frac{1}{12} C \left (-\log \left (a^{2/3}-2 \sqrt [3]{a} x+4 x^2\right )+\log \left (a+8 x^3\right )+2 \log \left (\sqrt [3]{a}+2 x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(-2*Sqrt[3]*ArcTan[(1 - (4*x)/a^(1/3))/Sqrt[3]] + 2*Log[a^(1/3) + 2*x] - Log[a^(2/3) - 2*a^(1/3)*x + 4*x^2]
+ Log[a + 8*x^3]))/12

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Maple [B]  time = 0.047, size = 84, normalized size = 1.8 \begin{align*}{\frac{C{8}^{{\frac{2}{3}}}}{24}\ln \left ( x+{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ) }-{\frac{C{8}^{{\frac{2}{3}}}}{48}\ln \left ({x}^{2}-{\frac{{8}^{{\frac{2}{3}}}x}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ) }+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}-1 \right ) } \right ) }+{\frac{C\ln \left ( 8\,{x}^{3}+a \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*C*8^(2/3)*ln(x+1/8*8^(2/3)*a^(1/3))-1/48*C*8^(2/3)*ln(x^2-1/8*8^(2/3)*a^(1/3)*x+1/8*8^(1/3)*a^(2/3))+1/24
*C*8^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*8^(1/3)/a^(1/3)*x-1))+1/12*C*ln(8*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((a^(2/3)*C+2*C*x^2)/(8*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01958, size = 122, normalized size = 2.6 \begin{align*} \frac{1}{6} \, \sqrt{3} C \arctan \left (\frac{4 \, \sqrt{3} a^{\frac{2}{3}} x - \sqrt{3} a}{3 \, a}\right ) + \frac{1}{4} \, C \log \left (2 \, x + a^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*C*arctan(1/3*(4*sqrt(3)*a^(2/3)*x - sqrt(3)*a)/a) + 1/4*C*log(2*x + a^(1/3))

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Sympy [C]  time = 0.442604, size = 85, normalized size = 1.81 \begin{align*} C \left (\frac{\log{\left (\frac{\sqrt [3]{a}}{2} + x \right )}}{4} - \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} - \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12} + \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} + \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*(log(a**(1/3)/2 + x)/4 - sqrt(3)*I*log(x + (-C*a**(1/3) - sqrt(3)*I*C*a**(1/3))/(4*C))/12 + sqrt(3)*I*log(x
+ (-C*a**(1/3) + sqrt(3)*I*C*a**(1/3))/(4*C))/12)

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Giac [B]  time = 1.09818, size = 150, normalized size = 3.19 \begin{align*} \frac{\sqrt{3}{\left (\sqrt{3} i{\left | a \right |} + a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{12 \, a} + \frac{{\left (\sqrt{3} i{\left | a \right |} + 3 \, a\right )} C \log \left (x^{2} + \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} x + \frac{1}{4} \, \left (-a\right )^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (C \left (-a\right )^{\frac{2}{3}} + 2 \, C a^{\frac{2}{3}}\right )} \left (-a\right )^{\frac{1}{3}} \log \left ({\left | x - \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} \right |}\right )}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*(sqrt(3)*i*abs(a) + a)*C*arctan(1/3*sqrt(3)*(4*x + (-a)^(1/3))/(-a)^(1/3))/a + 1/24*(sqrt(3)*i*ab
s(a) + 3*a)*C*log(x^2 + 1/2*(-a)^(1/3)*x + 1/4*(-a)^(2/3))/a - 1/12*(C*(-a)^(2/3) + 2*C*a^(2/3))*(-a)^(1/3)*lo
g(abs(x - 1/2*(-a)^(1/3)))/a