3.63 \(\int \frac{A+B x^3}{x \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=34 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log \left (a+b x^3\right )}{3 a b} \]

[Out]

(A*Log[x])/a - ((A*b - a*B)*Log[a + b*x^3])/(3*a*b)

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Rubi [A]  time = 0.100573, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log \left (a+b x^3\right )}{3 a b} \]

Antiderivative was successfully verified.

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

[Out]

(A*Log[x])/a - ((A*b - a*B)*Log[a + b*x^3])/(3*a*b)

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Rubi in Sympy [A]  time = 12.5917, size = 29, normalized size = 0.85 \[ \frac{A \log{\left (x^{3} \right )}}{3 a} - \frac{\left (A b - B a\right ) \log{\left (a + b x^{3} \right )}}{3 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x**3)/(3*a) - (A*b - B*a)*log(a + b*x**3)/(3*a*b)

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Mathematica [A]  time = 0.0237869, size = 34, normalized size = 1. \[ \frac{(a B-A b) \log \left (a+b x^3\right )}{3 a b}+\frac{A \log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

(A*Log[x])/a + ((-(A*b) + a*B)*Log[a + b*x^3])/(3*a*b)

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Maple [A]  time = 0.007, size = 37, normalized size = 1.1 \[{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,a}}+{\frac{\ln \left ( b{x}^{3}+a \right ) B}{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.376, size = 47, normalized size = 1.38 \[ \frac{A \log \left (x^{3}\right )}{3 \, a} + \frac{{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*A*log(x^3)/a + 1/3*(B*a - A*b)*log(b*x^3 + a)/(a*b)

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Fricas [A]  time = 0.231015, size = 43, normalized size = 1.26 \[ \frac{3 \, A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*A*b*log(x) + (B*a - A*b)*log(b*x^3 + a))/(a*b)

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Sympy [A]  time = 3.10654, size = 26, normalized size = 0.76 \[ \frac{A \log{\left (x \right )}}{a} + \frac{\left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/a + (-A*b + B*a)*log(a/b + x**3)/(3*a*b)

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GIAC/XCAS [A]  time = 0.216665, size = 46, normalized size = 1.35 \[ \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a} + \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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