3.75 \(\int \frac{x^5 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=60 \[ \frac{a (A b-a B)}{3 b^3 \left (a+b x^3\right )}+\frac{(A b-2 a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^3}{3 b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.178759, antiderivative size = 60, 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 \[ \frac{a (A b-a B)}{3 b^3 \left (a+b x^3\right )}+\frac{(A b-2 a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a \left (A b - B a\right )}{3 b^{3} \left (a + b x^{3}\right )} + \frac{\int ^{x^{3}} B\, dx}{3 b^{2}} + \frac{\left (A b - 2 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0661222, size = 50, normalized size = 0.83 \[ \frac{\frac{a (A b-a B)}{a+b x^3}+(A b-2 a B) \log \left (a+b x^3\right )+b B x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 74, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{2}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) Ba}{3\,{b}^{3}}}+{\frac{aA}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}B}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.37334, size = 81, normalized size = 1.35 \[ \frac{B x^{3}}{3 \, b^{2}} - \frac{B a^{2} - A a b}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.223549, size = 109, normalized size = 1.82 \[ \frac{B b^{2} x^{6} + B a b x^{3} - B a^{2} + A a b -{\left ({\left (2 \, B a b - A b^{2}\right )} x^{3} + 2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.83167, size = 56, normalized size = 0.93 \[ \frac{B x^{3}}{3 b^{2}} - \frac{- A a b + B a^{2}}{3 a b^{3} + 3 b^{4} x^{3}} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**3/(3*b**2) - (-A*a*b + B*a**2)/(3*a*b**3 + 3*b**4*x**3) - (-A*b + 2*B*a)*lo
g(a + b*x**3)/(3*b**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217873, size = 123, normalized size = 2.05 \[ \frac{\frac{{\left (b x^{3} + a\right )} B}{b^{2}} + \frac{{\left (2 \, B a - A b\right )}{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} - \frac{\frac{B a^{2} b}{b x^{3} + a} - \frac{A a b^{2}}{b x^{3} + a}}{b^{3}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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