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

Optimal. Leaf size=66 \[ -\frac{A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac{a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}+\frac{B \log \left (a+b x^3\right )}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.17726, antiderivative size = 66, 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 \[ -\frac{A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac{a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}+\frac{B \log \left (a+b x^3\right )}{3 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.4061, size = 56, normalized size = 0.85 \[ \frac{B \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{a \left (A b - B a\right )}{6 b^{3} \left (a + b x^{3}\right )^{2}} - \frac{A b - 2 B a}{3 b^{3} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0454318, size = 64, normalized size = 0.97 \[ \frac{3 a^2 B-a b \left (A-4 B x^3\right )+2 B \left (a+b x^3\right )^2 \log \left (a+b x^3\right )-2 A b^2 x^3}{6 b^3 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 81, normalized size = 1.2 \[{\frac{aA}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}B}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}}-{\frac{A}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,Ba}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.40649, size = 97, normalized size = 1.47 \[ \frac{2 \,{\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{B \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22667, size = 120, normalized size = 1.82 \[ \frac{2 \,{\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b + 2 \,{\left (B b^{2} x^{6} + 2 \, B a b x^{3} + B a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.11545, size = 70, normalized size = 1.06 \[ \frac{B \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{- A a b + 3 B a^{2} + x^{3} \left (- 2 A b^{2} + 4 B a b\right )}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219721, size = 82, normalized size = 1.24 \[ \frac{B{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x^{3} + \frac{3 \, B a^{2} - A a b}{b}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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