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

Optimal. Leaf size=54 \[ -\frac{a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.154003, antiderivative size = 54, 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 \[ -\frac{a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{3}} x\, dx}{3 b} - \frac{a \left (A b - B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \left (\frac{A b}{3} - \frac{B a}{3}\right ) \int ^{x^{3}} \frac{1}{b^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0378745, size = 47, normalized size = 0.87 \[ \frac{b x^3 \left (-2 a B+2 A b+b B x^3\right )+2 a (a B-A b) \log \left (a+b x^3\right )}{6 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 62, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,b}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{B{x}^{3}a}{3\,{b}^{2}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{2}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{3\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^3+A)/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.37947, size = 68, normalized size = 1.26 \[ \frac{B b x^{6} - 2 \,{\left (B a - A b\right )} x^{3}}{6 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232218, size = 69, normalized size = 1.28 \[ \frac{B b^{2} x^{6} - 2 \,{\left (B a b - A b^{2}\right )} x^{3} + 2 \,{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.07418, size = 44, normalized size = 0.81 \[ \frac{B x^{6}}{6 b} + \frac{a \left (- A b + B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219432, size = 70, normalized size = 1.3 \[ \frac{B b x^{6} - 2 \, B a x^{3} + 2 \, A b x^{3}}{6 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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