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

Optimal. Leaf size=35 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac{B x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.10055, antiderivative size = 35, 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 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac{B x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0221723, size = 31, normalized size = 0.89 \[ \frac{(A b-a B) \log \left (a+b x^3\right )+b B x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 1.1 \[{\frac{B{x}^{3}}{3\,b}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,b}}-{\frac{\ln \left ( b{x}^{3}+a \right ) Ba}{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x^3+A)/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.37096, size = 42, normalized size = 1.2 \[ \frac{B x^{3}}{3 \, b} - \frac{{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222246, size = 41, normalized size = 1.17 \[ \frac{B b x^{3} -{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.90815, size = 27, normalized size = 0.77 \[ \frac{B x^{3}}{3 b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216376, size = 43, normalized size = 1.23 \[ \frac{B x^{3}}{3 \, b} - \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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