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

Optimal. Leaf size=21 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

[Out]

(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

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Rubi [A]  time = 0.0333973, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

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Rubi in Sympy [A]  time = 16.0681, size = 19, normalized size = 0.9 \[ \frac{C \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{\sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**(2/3)*C-a**(1/3)*b**(1/3)*C*x+b**(2/3)*C*x**2)/(b*x**3+a),x)

[Out]

C*log(a**(1/3) + b**(1/3)*x)/b**(1/3)

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Mathematica [A]  time = 0.00519268, size = 21, normalized size = 1. \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)

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Maple [B]  time = 0.007, size = 218, normalized size = 10.4 \[{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{6\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256511, size = 23, normalized size = 1.1 \[ \frac{C \log \left (b x + a^{\frac{1}{3}} b^{\frac{2}{3}}\right )}{b^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*log(b*x + a^(1/3)*b^(2/3))/b^(1/3)

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Sympy [A]  time = 0.623161, size = 20, normalized size = 0.95 \[ \frac{C \log{\left (\sqrt [3]{a} b^{\frac{2}{3}} + b x \right )}}{\sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**(2/3)*C-a**(1/3)*b**(1/3)*C*x+b**(2/3)*C*x**2)/(b*x**3+a),x)

[Out]

C*log(a**(1/3)*b**(2/3) + b*x)/b**(1/3)

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GIAC/XCAS [A]  time = 0.223282, size = 22, normalized size = 1.05 \[ \frac{C{\rm ln}\left ({\left | b^{\frac{1}{3}} x + a^{\frac{1}{3}} \right |}\right )}{b^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*ln(abs(b^(1/3)*x + a^(1/3)))/b^(1/3)