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)

_______________________________________________________________________________________

Rubi [A]  time = 0.185266, 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 \[ \frac{2 \left (\sqrt [3]{a} (-b)^{2/3} C+b B\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b}+\frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.39, size = 78, normalized size = 0.89 \[ \frac{C \log{\left (\sqrt [3]{a} - x \sqrt [3]{- b} \right )}}{\sqrt [3]{- b}} + \frac{2 \sqrt{3} \left (\frac{B}{\sqrt [3]{a}} + \frac{C \left (- b\right )^{\frac{2}{3}}}{b}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{- b}}{3}\right )}{\sqrt [3]{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

C*log(a**(1/3) - x*(-b)**(1/3))/(-b)**(1/3) + 2*sqrt(3)*(B/a**(1/3) + C*(-b)**(2
/3)/b)*atan(sqrt(3)*(a**(1/3)/3 + 2*x*(-b)**(1/3)/3)/a**(1/3))/3

_______________________________________________________________________________________

Mathematica [B]  time = 1.23865, 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))/Sqrt[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)

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 345, normalized size = 3.9 \[{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \sqrt [3]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \sqrt [3]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \sqrt [3]{a}\sqrt [3]{-b} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{6\,b} \left ( -b \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \]

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/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^(1/3)*(-b)^(1/3)*B-2/3*C*a^(2/3)/b/(a/b)^(
2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^(1/
3)*(-b)^(1/3)*B+1/3*C*a^(2/3)/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/
3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*a^(1/3)*(-b)^(1/
3)*B-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*B*(-b)^(2/3)/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6*B*(-b)^(2/3)/b/(a/b)^(1/3)
*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*B*(-b)^(2/3)*3^(1/2)/b/(a/b)^(1/3)*arctan
(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3*C*(-b)^(2/3)/b*ln(b*x^3+a)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out