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

Optimal. Leaf size=76 \[ \frac{2 \left (C \sqrt [3]{-\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.102125, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {1867, 31, 617, 204} \[ \frac{2 \left (C \sqrt [3]{-\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

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*(B + (-(a/b))^(1/3)*C)*ArcTan[(1 - (2*x)/(-(a/b))^(1/3))/Sqrt[3]])/(Sqrt[3]*(-(a/b))^(1/3)*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{\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}-\frac{\left (2 \left (\frac{B}{\sqrt [3]{-\frac{a}{b}}}+C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}\right )}{b}\\ &=\frac{2 \left (\frac{B}{\sqrt [3]{-\frac{a}{b}}}+C\right ) \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 = 0.177006, size = 288, normalized size = 3.79 \[ -\frac{\left (-a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}-2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{2/3}}-\frac{\left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}+2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a b^{2/3}}-\frac{\left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}-2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a b^{2/3}}-\frac{C \log \left (a-b x^3\right )}{3 b} \]

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]

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

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Maple [B]  time = 0.006, size = 345, normalized size = 4.5 \begin{align*} -{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \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*(-1/b*a)^(2/3)/b/(1/b*a)^(2/3)*ln(x-(1/b*a)^(1/3))-1/3/b/(1/b*a)^(2/3)*ln(x-(1/b*a)^(1/3))*(-1/b*a)^(1/
3)*B+1/3*C*(-1/b*a)^(2/3)/b/(1/b*a)^(2/3)*ln(x^2+(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/6/b/(1/b*a)^(2/3)*ln(x^2+(1/
b*a)^(1/3)*x+(1/b*a)^(2/3))*(-1/b*a)^(1/3)*B+2/3*C*(-1/b*a)^(2/3)/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*(1+2/(1/b
*a)^(1/3)*x)*3^(1/2))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*(1+2/(1/b*a)^(1/3)*x)*3^(1/2))*(-1/b*a)^(1/3)*B-1
/3*B/b/(1/b*a)^(1/3)*ln(x-(1/b*a)^(1/3))+1/6*B/b/(1/b*a)^(1/3)*ln(x^2+(1/b*a)^(1/3)*x+(1/b*a)^(2/3))-1/3*B*3^(
1/2)/b/(1/b*a)^(1/3)*arctan(1/3*(1+2/(1/b*a)^(1/3)*x)*3^(1/2))-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(((-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

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Fricas [B]  time = 5.77571, size = 972, normalized size = 12.79 \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]

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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

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Giac [B]  time = 1.10247, size = 351, normalized size = 4.62 \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)