3.66 \(\int \frac{A+B x^3}{x^4 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=50 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.138746, antiderivative size = 50, 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 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^4*(a + b*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.4592, size = 44, normalized size = 0.88 \[ - \frac{A}{3 a x^{3}} - \frac{\left (A b - B a\right ) \log{\left (x^{3} \right )}}{3 a^{2}} + \frac{\left (A b - B a\right ) \log{\left (a + b x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**4/(b*x**3+a),x)

[Out]

-A/(3*a*x**3) - (A*b - B*a)*log(x**3)/(3*a**2) + (A*b - B*a)*log(a + b*x**3)/(3*
a**2)

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Mathematica [A]  time = 0.0380239, size = 49, normalized size = 0.98 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 a^2}+\frac{\log (x) (a B-A b)}{a^2}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^4*(a + b*x^3)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 56, normalized size = 1.1 \[ -{\frac{A}{3\,a{x}^{3}}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{B\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( b{x}^{3}+a \right ) Ab}{3\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) B}{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^4/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.38832, size = 65, normalized size = 1.3 \[ -\frac{{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2}} + \frac{{\left (B a - A b\right )} \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{A}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^4),x, algorithm="maxima")

[Out]

-1/3*(B*a - A*b)*log(b*x^3 + a)/a^2 + 1/3*(B*a - A*b)*log(x^3)/a^2 - 1/3*A/(a*x^
3)

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Fricas [A]  time = 0.228558, size = 63, normalized size = 1.26 \[ -\frac{{\left (B a - A b\right )} x^{3} \log \left (b x^{3} + a\right ) - 3 \,{\left (B a - A b\right )} x^{3} \log \left (x\right ) + A a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^4),x, algorithm="fricas")

[Out]

-1/3*((B*a - A*b)*x^3*log(b*x^3 + a) - 3*(B*a - A*b)*x^3*log(x) + A*a)/(a^2*x^3)

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Sympy [A]  time = 3.81643, size = 41, normalized size = 0.82 \[ - \frac{A}{3 a x^{3}} + \frac{\left (- A b + B a\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**4/(b*x**3+a),x)

[Out]

-A/(3*a*x**3) + (-A*b + B*a)*log(x)/a**2 - (-A*b + B*a)*log(a/b + x**3)/(3*a**2)

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GIAC/XCAS [A]  time = 0.217731, size = 93, normalized size = 1.86 \[ \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (B a b - A b^{2}\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b} - \frac{B a x^{3} - A b x^{3} + A a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^4),x, algorithm="giac")

[Out]

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