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

Optimal. Leaf size=61 \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {2 C \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1863, 31, 617, 204} \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {2 C \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

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])

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 95, normalized size = 1.56 \[ \frac {C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )\right )}{3 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 160, normalized size = 2.62 \[ \left [\frac {\sqrt {\frac {1}{3}} C b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} - 3 \, a^{\frac {2}{3}} b^{\frac {1}{3}} x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {1}{3}} b x^{2} + a^{\frac {2}{3}} b^{\frac {2}{3}} x - a b^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} - a}{b x^{3} + a}\right ) + C b^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} b^{\frac {2}{3}}\right )}{b}, \frac {2 \, \sqrt {\frac {1}{3}} C b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a^{\frac {2}{3}} b^{\frac {2}{3}} x - a b^{\frac {1}{3}}\right )}}{a b^{\frac {1}{3}}}\right ) + C b^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} b^{\frac {2}{3}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 117, normalized size = 1.92 \[ \frac {2 \sqrt {3}\, C \,a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {2 C \,a^{\frac {2}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {C \,a^{\frac {2}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {C \ln \left (b \,x^{3}+a \right )}{3 b^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 2.87, size = 162, normalized size = 2.66 \[ -\frac {2 \, \sqrt {3} {\left (C a b^{\frac {2}{3}} - {\left (3 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {C a}{b^{\frac {1}{3}}}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - C a^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C a^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.31, size = 193, normalized size = 3.16 \[ \sum _{k=1}^3\ln \left (-\frac {a^{2/3}\,\left (C-\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^{8/3}\,z^2+9\,C^2\,a^2\,b^{7/3}\,z-9\,C^3\,a^2\,b^2,z,k\right )\,b^{1/3}\,3\right )\,\left (-C\,a^{1/3}+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^{8/3}\,z^2+9\,C^2\,a^2\,b^{7/3}\,z-9\,C^3\,a^2\,b^2,z,k\right )\,a^{1/3}\,b^{1/3}\,3+2\,C\,b^{1/3}\,x\right )}{b^{5/3}}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^{8/3}\,z^2+9\,C^2\,a^2\,b^{7/3}\,z-9\,C^3\,a^2\,b^2,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-(a^(2/3)*(C - 3*root(27*a^2*b^3*z^3 - 27*C*a^2*b^(8/3)*z^2 + 9*C^2*a^2*b^(7/3)*z - 9*C^3*a^2*b^2,
z, k)*b^(1/3))*(3*root(27*a^2*b^3*z^3 - 27*C*a^2*b^(8/3)*z^2 + 9*C^2*a^2*b^(7/3)*z - 9*C^3*a^2*b^2, z, k)*a^(1
/3)*b^(1/3) - C*a^(1/3) + 2*C*b^(1/3)*x))/b^(5/3))*root(27*a^2*b^3*z^3 - 27*C*a^2*b^(8/3)*z^2 + 9*C^2*a^2*b^(7
/3)*z - 9*C^3*a^2*b^2, z, k), k, 1, 3)

________________________________________________________________________________________

sympy [A]  time = 0.73, size = 70, normalized size = 1.15 \[ \operatorname {RootSum} {\left (3 t^{3} b^{\frac {5}{3}} - 3 t^{2} C b^{\frac {4}{3}} + t C^{2} b - C^{3} b^{\frac {2}{3}}, \left (t \mapsto t \log {\left (x + \frac {3 t \sqrt [3]{a} \sqrt [3]{b} - C \sqrt [3]{a}}{2 C \sqrt [3]{b}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a**(2/3)*C+b**(2/3)*C*x**2)/(b*x**3+a),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*a**(1/3
)*b**(1/3) - C*a**(1/3))/(2*C*b**(1/3)))))

________________________________________________________________________________________