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

Optimal. Leaf size=78 \[ \frac{2 \left (B-C \sqrt [3]{-\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b} \]

[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 = 0.107623, antiderivative size = 78, 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 = {1869, 31, 617, 204} \[ \frac{2 \left (B-C \sqrt [3]{-\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b} \]

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 1869

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 (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 )}{\sqrt [3]{-\frac{a}{b}} b}\\ &=\frac{2 \left (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} \sqrt [3]{-\frac{a}{b}} b}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}\\ \end{align*}

Mathematica [B]  time = 0.251451, size = 253, normalized size = 3.24 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (B-2 C \sqrt [3]{-\frac{a}{b}}\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (B-2 C \sqrt [3]{-\frac{a}{b}}\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (2 C \sqrt [3]{-\frac{a}{b}}-B\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]

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))*Lo
g[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)

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Maple [B]  time = 0.005, size = 340, normalized size = 4.4 \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 ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \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 ( 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*(-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*3^(1/2)*(2
/(1/b*a)^(1/3)*x-1))-1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*(-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*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((-(-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 = 5.75111, size = 968, normalized size = 12.41 \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 [A]  time = 1.12964, size = 180, normalized size = 2.31 \begin{align*} -\frac{2 \, \sqrt{3}{\left (C a b + \left (-a b^{2}\right )^{\frac{2}{3}} 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 )}{3 \, a b^{2}} - \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}} \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]

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