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

Optimal. Leaf size=71 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0941567, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1867, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]

Antiderivative was successfully verified.

[In]

Int[((a/b)^(1/3)*B + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]

[Out]

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

Mathematica [B]  time = 0.267454, size = 247, normalized size = 3.48 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )-a^{2/3} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]

Antiderivative was successfully verified.

[In]

Integrate[((a/b)^(1/3)*B + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]

[Out]

(2*Sqrt[3]*a^(1/3)*b^(1/3)*(a^(1/3)*B + (a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*ArcTan[(-a^(1/3) + 2*b^(1/3
)*x)/(Sqrt[3]*a^(1/3))] + 2*b^(1/3)*(-(a^(2/3)*B) + a^(1/3)*(a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*Log[a^(
1/3) + b^(1/3)*x] + b^(1/3)*(a^(2/3)*B - a^(1/3)*(a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*Log[a^(2/3) - a^(1
/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*C*Log[a + b*x^3])/(6*a*b)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 121, 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{2\,B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\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(((1/b*a)^(1/3)*B+2*(1/b*a)^(2/3)*C+B*x+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))+2/3*B*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/3*C/b*ln(b*x^3
+a)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 5.69123, size = 948, normalized size = 13.35 \begin{align*} \left [\frac{C \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + \sqrt{\frac{1}{3}} \sqrt{-\frac{2 \, B C b \left (\frac{a}{b}\right )^{\frac{2}{3}} + B^{2} b \left (\frac{a}{b}\right )^{\frac{1}{3}} + C^{2} a}{a}} \log \left (-\frac{C^{3} a^{2} + B^{3} a b - 2 \,{\left (C^{3} a b + B^{3} b^{2}\right )} x^{3} + 3 \,{\left (C^{3} a b + B^{3} b^{2}\right )} x \left (\frac{a}{b}\right )^{\frac{2}{3}} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, B C a b x^{2} - B^{2} a b x + C^{2} a^{2} -{\left (2 \, B^{2} b^{2} x^{2} + C^{2} a b x + B C a b\right )} \left (\frac{a}{b}\right )^{\frac{2}{3}} -{\left (2 \, C^{2} a b x^{2} - B C a b x - B^{2} a b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{2 \, B C b \left (\frac{a}{b}\right )^{\frac{2}{3}} + B^{2} b \left (\frac{a}{b}\right )^{\frac{1}{3}} + C^{2} a}{a}}}{b x^{3} + a}\right )}{b}, \frac{2 \, \sqrt{\frac{1}{3}} \sqrt{\frac{2 \, B C b \left (\frac{a}{b}\right )^{\frac{2}{3}} + B^{2} b \left (\frac{a}{b}\right )^{\frac{1}{3}} + C^{2} a}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, B^{2} b x - C^{2} a +{\left (2 \, C^{2} b x + B C b\right )} \left (\frac{a}{b}\right )^{\frac{2}{3}} -{\left (2 \, B C b x + B^{2} b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )} \sqrt{\frac{2 \, B C b \left (\frac{a}{b}\right )^{\frac{2}{3}} + B^{2} b \left (\frac{a}{b}\right )^{\frac{1}{3}} + C^{2} a}{a}}}{C^{3} a + B^{3} b}\right ) + C \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+C*x^2)/(b*x^3+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialDivisionFailed} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialDivisionFailed

________________________________________________________________________________________

Giac [B]  time = 1.2488, size = 362, normalized size = 5.1 \begin{align*} -\frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (a b^{2}\right )^{\frac{1}{3}} B b + 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{\sqrt{3}{\left ({\left (9 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} - 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B + 18 \,{\left (a^{2} b^{3} - \sqrt{3} \sqrt{a^{4} b^{6}} i\right )} C\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 )}{54 \, a^{2} b^{4}} + \frac{{\left ({\left (27 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} + 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B + 18 \,{\left (3 \, a^{2} b^{3} - \sqrt{3} \sqrt{a^{4} b^{6}} i\right )} C\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{108 \, a^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+C*x^2)/(b*x^3+a),x, algorithm="giac")

[Out]

-1/3*(C*b^2*(-a/b)^(2/3) + B*b^2*(-a/b)^(1/3) + (a*b^2)^(1/3)*B*b + 2*(a*b^2)^(2/3)*C)*(-a/b)^(1/3)*log(abs(x
- (-a/b)^(1/3)))/(a*b^2) + 1/54*sqrt(3)*((9*(-a^2*b^4)^(1/3)*a*b^2 - 27^(5/6)*(-a^2*b^4)^(5/6))*B + 18*(a^2*b^
3 - sqrt(3)*sqrt(a^4*b^6)*i)*C)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) + 1/108*((27*(
-a^2*b^4)^(1/3)*a*b^2 + 27^(5/6)*(-a^2*b^4)^(5/6))*B + 18*(3*a^2*b^3 - sqrt(3)*sqrt(a^4*b^6)*i)*C)*log(x^2 + x
*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^4)