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

Optimal. Leaf size=88 \[ \frac{2 \left (\sqrt [3]{a} (-b)^{2/3} C+b B\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b}+\frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \]

[Out]

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

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Rubi [A]  time = 0.112343, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 57, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.07, Rules used = {1866, 31, 617, 204} \[ \frac{2 \left (\sqrt [3]{a} (-b)^{2/3} C+b B\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b}+\frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1866

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]

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]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx &=-\frac{C \int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{-b}}-x} \, dx}{\sqrt [3]{-b}}+\frac{\left (\sqrt [3]{-b} B-\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 \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\left (2 \left (\frac{B}{\sqrt [3]{a}}+\frac{b C}{(-b)^{4/3}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{-b} x}{\sqrt [3]{a}}\right )\\ &=\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{b C}{(-b)^{4/3}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}}+\frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}\\ \end{align*}

Mathematica [B]  time = 0.526485, size = 238, normalized size = 2.7 \[ \frac{\frac{\left (2 \sqrt [3]{a} b \sqrt [3]{-b} C+b^{5/3} B+(-b)^{5/3} B\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 b \left (2 \sqrt [3]{a} \sqrt [3]{-b} C+\left (b^{2/3}-(-b)^{2/3}\right ) B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt [3]{a} (-b)^{2/3} \sqrt [3]{-b^2} C \log \left (a+b x^3\right )}{\sqrt [3]{-b^2}}+2 \sqrt{3} \sqrt [3]{b} \left (2 \sqrt [3]{a} \sqrt [3]{b} C+\left ((-b)^{2/3}-\sqrt [3]{-b^2}\right ) B\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 \sqrt [3]{a} b} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[3]*b^(1/3)*(((-b)^(2/3) - (-b^2)^(1/3))*B + 2*a^(1/3)*b^(1/3)*C)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sq
rt[3]] + (-2*b*((-(-b)^(2/3) + b^(2/3))*B + 2*a^(1/3)*(-b)^(1/3)*C)*Log[a^(1/3) + b^(1/3)*x] + ((-b)^(5/3)*B +
b^(5/3)*B + 2*a^(1/3)*(-b)^(1/3)*b*C)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a^(1/3)*(-b)^(2/3)*(
-b^2)^(1/3)*C*Log[a + b*x^3])/(-b^2)^(1/3))/(6*a^(1/3)*b)

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Maple [B]  time = 0.006, size = 345, normalized size = 3.9 \begin{align*}{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \sqrt [3]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \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]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b}{a}^{{\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{\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]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\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{B}{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{6\,b} \left ( -b \right ) ^{{\frac{2}{3}}}\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} \left ( -b \right ) ^{{\frac{2}{3}}}\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} \left ( -b \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^2)/(b*x^3+a),x)

[Out]

1/3/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a^(1/3)*(-b)^(1/3)*B-2/3*C*a^(2/3)/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))
-1/6/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*a^(1/3)*(-b)^(1/3)*B+1/3*C*a^(2/3)/b/(1/b*a)^(2/3)*
ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+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))*a^
(1/3)*(-b)^(1/3)*B-2/3*C*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*(-b)^
(2/3)/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-1/6*B*(-b)^(2/3)/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3
))-1/3*B*(-b)^(2/3)*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)^(2/3)/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^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^2)/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 9.57486, size = 1129, normalized size = 12.83 \begin{align*} \left [\frac{\sqrt{\frac{1}{3}} b \sqrt{\frac{C^{2} a \left (-b\right )^{\frac{1}{3}} - 2 \, B C a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} - B^{2} a^{\frac{1}{3}} b}{a b}} \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^{3} b\right )} a^{\frac{2}{3}} \left (-b\right )^{\frac{1}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left ({\left (2 \, B^{2} b x^{2} + C^{2} a x + B C a\right )} a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} +{\left (2 \, C^{2} a b x^{2} - B C a b x - B^{2} a b\right )} a^{\frac{1}{3}} +{\left (2 \, B C a b x^{2} - B^{2} a b x + C^{2} a^{2}\right )} \left (-b\right )^{\frac{1}{3}}\right )} \sqrt{\frac{C^{2} a \left (-b\right )^{\frac{1}{3}} - 2 \, B C a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} - B^{2} a^{\frac{1}{3}} b}{a b}}}{b x^{3} + a}\right ) - C \left (-b\right )^{\frac{2}{3}} \log \left (b x + a^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} b \sqrt{-\frac{C^{2} a \left (-b\right )^{\frac{1}{3}} - 2 \, B C a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} - B^{2} a^{\frac{1}{3}} b}{a b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left ({\left (2 \, C^{2} x + B C\right )} a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} -{\left (2 \, B C b x + B^{2} b\right )} a^{\frac{1}{3}} -{\left (2 \, B^{2} b x - C^{2} a\right )} \left (-b\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{C^{2} a \left (-b\right )^{\frac{1}{3}} - 2 \, B C a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} - B^{2} a^{\frac{1}{3}} b}{a b}}}{C^{3} a + B^{3} b}\right ) + C \left (-b\right )^{\frac{2}{3}} \log \left (b x + a^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: PolynomialDivisionFailed

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out