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

Optimal. Leaf size=50 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]

[Out]

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

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Rubi [A]  time = 0.0765061, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1867, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1867

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

Mathematica [B]  time = 0.0519632, size = 146, normalized size = 2.92 \[ \frac{C \left (-b^{2/3} \left (\frac{a}{b}\right )^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )+2 b^{2/3} \left (\frac{a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \left (\frac{a}{b}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 87, normalized size = 1.7 \begin{align*}{\frac{2\,C}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{C}{3\,b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) }+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(1/b*a)^(2/3)*C+C*x^2)/(b*x^3+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04965, size = 138, normalized size = 2.76 \begin{align*} \frac{2 \, \sqrt{3} C \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 3 \, C \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 0.536165, size = 100, normalized size = 2. \begin{align*} \frac{C \left (\log{\left (\frac{a}{b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} - \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*(log(a/(b*(a/b)**(2/3)) + x) - sqrt(3)*I*log(-a/(2*b*(a/b)**(2/3)) - sqrt(3)*I*a/(2*b*(a/b)**(2/3)) + x)/3 +
sqrt(3)*I*log(-a/(2*b*(a/b)**(2/3)) + sqrt(3)*I*a/(2*b*(a/b)**(2/3)) + x)/3)/b

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Giac [B]  time = 1.12556, size = 224, normalized size = 4.48 \begin{align*} \frac{\sqrt{3}{\left (a b^{2} + \sqrt{3} \sqrt{a^{2} b^{4}} i\right )} C \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 )}{3 \, a b^{3}} - \frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} + \frac{{\left (3 \, a b^{2} + \sqrt{3} \sqrt{a^{2} b^{4}} i\right )} C \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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