3.24 \(\int \left (-\frac{C x^2}{a+b x^3}+\frac{A+B x+C x^2}{a+b x^3}\right ) \, dx\)

Optimal. Leaf size=161 \[ -\frac{\left (A-\frac{\sqrt [3]{a} B}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} B+A \sqrt [3]{b}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.306013, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ -\frac{\left (A-\frac{\sqrt [3]{a} B}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} B+A \sqrt [3]{b}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.0368, size = 150, normalized size = 0.93 \[ \frac{\left (A \sqrt [3]{b} - B \sqrt [3]{a}\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}}} - \frac{\left (A \sqrt [3]{b} - B \sqrt [3]{a}\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (A \sqrt [3]{b} + B \sqrt [3]{a}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-C*x**2/(b*x**3+a)+(C*x**2+B*x+A)/(b*x**3+a),x)

[Out]

(A*b**(1/3) - B*a**(1/3))*log(a**(1/3) + b**(1/3)*x)/(3*a**(2/3)*b**(2/3)) - (A*
b**(1/3) - B*a**(1/3))*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a*
*(2/3)*b**(2/3)) - sqrt(3)*(A*b**(1/3) + B*a**(1/3))*atan(sqrt(3)*(a**(1/3)/3 -
2*b**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)*b**(2/3))

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Mathematica [A]  time = 0.0731088, size = 124, normalized size = 0.77 \[ \frac{\left (A \sqrt [3]{b}-\sqrt [3]{a} B\right ) \left (2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )-2 \sqrt{3} \left (\sqrt [3]{a} B+A \sqrt [3]{b}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} b^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 186, normalized size = 1.2 \[{\frac{A}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-C*x^2/(b*x^3+a)+(C*x^2+B*x+A)/(b*x^3+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.12174, size = 76, normalized size = 0.47 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{2} + 9 t A B a b - A^{3} b + B^{3} a, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} B a^{2} b + 3 t A^{2} a b + 2 A B^{2} a}{A^{3} b + B^{3} a} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-C*x**2/(b*x**3+a)+(C*x**2+B*x+A)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**2*b**2 + 9*_t*A*B*a*b - A**3*b + B**3*a, Lambda(_t, _t*log(x
+ (9*_t**2*B*a**2*b + 3*_t*A**2*a*b + 2*A*B**2*a)/(A**3*b + B**3*a))))

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GIAC/XCAS [A]  time = 0.219074, size = 224, normalized size = 1.39 \[ -\frac{{\left (B b \left (-\frac{a}{b}\right )^{\frac{1}{3}} + A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} A b - \left (-a b^{2}\right )^{\frac{2}{3}} B\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} A a b^{3} + \left (-a b^{2}\right )^{\frac{2}{3}} B a b^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(B*b*(-a/b)^(1/3) + A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b) + 1/3
*sqrt(3)*((-a*b^2)^(1/3)*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) + 1/6*((-a*b^2)^(1/3)*A*a*b^3 + (-a*b^2)^(2/3)*B*
a*b^2)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^4)