\(\int \frac {2 (-\frac {a}{b})^{2/3} C+C x^2}{a+b x^3} \, dx\) [35]

Optimal result

Integrand size = 29, antiderivative size = 54 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=-\frac {2 C \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1883, 31, 631, 210} \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}-\frac {2 C \arctan \left (\frac {\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b} \]

[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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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 1883

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]

Rubi steps \begin{align*} \text {integral}& = -\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) \text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(54)=108\).

Time = 0.05 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.76 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \left (-2 \sqrt {3} \left (-\frac {a}{b}\right )^{2/3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (-\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\left (-\frac {a}{b}\right )^{2/3} b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )\right )}{3 a^{2/3} b} \]

[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)*L
og[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(47)=94\).

Time = 1.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.17

method	result	size
default	\(C \left (2 \left (-\frac {a}{b}\right )^{\frac {2}{3}} \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\right )\)	\(117\)

[In]

int((2*(-a/b)^(2/3)*C+C*x^2)/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 3 \, C \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.02 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\frac {C \left (\log {\left (\frac {a}{b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} - \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (47) = 94\).

Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 3.11 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} {\left (C a - {\left (3 \, C \left (\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + \frac {C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} - C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

-2/9*sqrt(3)*(C*a - (3*C*(a/b)^(1/3)*(-a/b)^(2/3) + C*a/b)*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/
3))/(a*b) + 1/3*(C*(a/b)^(2/3) - C*(-a/b)^(2/3))*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) + 1/3*
(C*(a/b)^(2/3) + 2*C*(-a/b)^(2/3))*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.69 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {{\left (C b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} C\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.38 (sec) , antiderivative size = 173, normalized size of antiderivative = 3.20 \[ \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (-\frac {\left (C-\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,b\,3\right )\,\left (-C\,a+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,a\,b\,3+2\,C\,b\,x\,{\left (-\frac {a}{b}\right )}^{2/3}\right )}{b^3}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right ) \]

[In]

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

[Out]

symsum(log(-((C - 3*root(27*a^2*b^3*z^3 - 27*C*a^2*b^2*z^2 + 9*C^2*a^2*b*z - 9*C^3*a^2, z, k)*b)*(3*root(27*a^
2*b^3*z^3 - 27*C*a^2*b^2*z^2 + 9*C^2*a^2*b*z - 9*C^3*a^2, z, k)*a*b - C*a + 2*C*b*x*(-a/b)^(2/3)))/b^3)*root(2
7*a^2*b^3*z^3 - 27*C*a^2*b^2*z^2 + 9*C^2*a^2*b*z - 9*C^3*a^2, z, k), k, 1, 3)