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

Optimal. Leaf size=101 \[ \frac{(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac{A}{3 a^3 x^3}-\frac{A b-a B}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

-A/(3*a^3*x^3) - (A*b - a*B)/(6*a^2*(a + b*x^3)^2) - (2*A*b - a*B)/(3*a^3*(a + b
*x^3)) - ((3*A*b - a*B)*Log[x])/a^4 + ((3*A*b - a*B)*Log[a + b*x^3])/(3*a^4)

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Rubi [A]  time = 0.268282, antiderivative size = 101, 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 \[ \frac{(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac{A}{3 a^3 x^3}-\frac{A b-a B}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a^3*x^3) - (A*b - a*B)/(6*a^2*(a + b*x^3)^2) - (2*A*b - a*B)/(3*a^3*(a + b
*x^3)) - ((3*A*b - a*B)*Log[x])/a^4 + ((3*A*b - a*B)*Log[a + b*x^3])/(3*a^4)

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Rubi in Sympy [A]  time = 23.4111, size = 90, normalized size = 0.89 \[ - \frac{A}{3 a^{3} x^{3}} - \frac{A b - B a}{6 a^{2} \left (a + b x^{3}\right )^{2}} - \frac{2 A b - B a}{3 a^{3} \left (a + b x^{3}\right )} - \frac{\left (3 A b - B a\right ) \log{\left (x^{3} \right )}}{3 a^{4}} + \frac{\left (3 A b - B a\right ) \log{\left (a + b x^{3} \right )}}{3 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*a**3*x**3) - (A*b - B*a)/(6*a**2*(a + b*x**3)**2) - (2*A*b - B*a)/(3*a**3*
(a + b*x**3)) - (3*A*b - B*a)*log(x**3)/(3*a**4) + (3*A*b - B*a)*log(a + b*x**3)
/(3*a**4)

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Mathematica [A]  time = 0.110146, size = 87, normalized size = 0.86 \[ \frac{\frac{a^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac{2 a (a B-2 A b)}{a+b x^3}+2 (3 A b-a B) \log \left (a+b x^3\right )+6 \log (x) (a B-3 A b)-\frac{2 a A}{x^3}}{6 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a*A)/x^3 + (a^2*(-(A*b) + a*B))/(a + b*x^3)^2 + (2*a*(-2*A*b + a*B))/(a + b
*x^3) + 6*(-3*A*b + a*B)*Log[x] + 2*(3*A*b - a*B)*Log[a + b*x^3])/(6*a^4)

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Maple [A]  time = 0.016, size = 117, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{3}{x}^{3}}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{B\ln \left ( x \right ) }{{a}^{3}}}-{\frac{Ab}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) A}{{a}^{4}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) B}{3\,{a}^{3}}}-{\frac{2\,Ab}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{B}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37408, size = 147, normalized size = 1.46 \[ \frac{2 \,{\left (B a b - 3 \, A b^{2}\right )} x^{6} + 3 \,{\left (B a^{2} - 3 \, A a b\right )} x^{3} - 2 \, A a^{2}}{6 \,{\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac{{\left (B a - 3 \, A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} + \frac{{\left (B a - 3 \, A b\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*(B*a*b - 3*A*b^2)*x^6 + 3*(B*a^2 - 3*A*a*b)*x^3 - 2*A*a^2)/(a^3*b^2*x^9 +
2*a^4*b*x^6 + a^5*x^3) - 1/3*(B*a - 3*A*b)*log(b*x^3 + a)/a^4 + 1/3*(B*a - 3*A*
b)*log(x^3)/a^4

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Fricas [A]  time = 0.227793, size = 266, normalized size = 2.63 \[ \frac{2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} - 2 \, A a^{3} + 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3} - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*(B*a^2*b - 3*A*a*b^2)*x^6 - 2*A*a^3 + 3*(B*a^3 - 3*A*a^2*b)*x^3 - 2*((B*a
*b^2 - 3*A*b^3)*x^9 + 2*(B*a^2*b - 3*A*a*b^2)*x^6 + (B*a^3 - 3*A*a^2*b)*x^3)*log
(b*x^3 + a) + 6*((B*a*b^2 - 3*A*b^3)*x^9 + 2*(B*a^2*b - 3*A*a*b^2)*x^6 + (B*a^3
- 3*A*a^2*b)*x^3)*log(x))/(a^4*b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)

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Sympy [A]  time = 13.358, size = 107, normalized size = 1.06 \[ \frac{- 2 A a^{2} + x^{6} \left (- 6 A b^{2} + 2 B a b\right ) + x^{3} \left (- 9 A a b + 3 B a^{2}\right )}{6 a^{5} x^{3} + 12 a^{4} b x^{6} + 6 a^{3} b^{2} x^{9}} + \frac{\left (- 3 A b + B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{\left (- 3 A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*A*a**2 + x**6*(-6*A*b**2 + 2*B*a*b) + x**3*(-9*A*a*b + 3*B*a**2))/(6*a**5*x*
*3 + 12*a**4*b*x**6 + 6*a**3*b**2*x**9) + (-3*A*b + B*a)*log(x)/a**4 - (-3*A*b +
B*a)*log(a/b + x**3)/(3*a**4)

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GIAC/XCAS [A]  time = 0.218881, size = 184, normalized size = 1.82 \[ \frac{{\left (B a - 3 \, A b\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b - 3 \, A b^{2}\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac{3 \, B a b^{2} x^{6} - 9 \, A b^{3} x^{6} + 8 \, B a^{2} b x^{3} - 22 \, A a b^{2} x^{3} + 6 \, B a^{3} - 14 \, A a^{2} b}{6 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{B a x^{3} - 3 \, A b x^{3} + A a}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*a - 3*A*b)*ln(abs(x))/a^4 - 1/3*(B*a*b - 3*A*b^2)*ln(abs(b*x^3 + a))/(a^4*b)
+ 1/6*(3*B*a*b^2*x^6 - 9*A*b^3*x^6 + 8*B*a^2*b*x^3 - 22*A*a*b^2*x^3 + 6*B*a^3 -
14*A*a^2*b)/((b*x^3 + a)^2*a^4) - 1/3*(B*a*x^3 - 3*A*b*x^3 + A*a)/(a^4*x^3)