3.89 \(\int \frac{x^8 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac{a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.25443, antiderivative size = 88, 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 \[ -\frac{a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac{a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (A b - B a\right )}{6 b^{4} \left (a + b x^{3}\right )^{2}} + \frac{a \left (2 A b - 3 B a\right )}{3 b^{4} \left (a + b x^{3}\right )} + \frac{\int ^{x^{3}} B\, dx}{3 b^{3}} + \frac{\left (A b - 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0681161, size = 92, normalized size = 1.05 \[ \frac{2 a A b-3 a^2 B}{3 b^4 \left (a+b x^3\right )}+\frac{a^3 B-a^2 A b}{6 b^4 \left (a+b x^3\right )^2}+\frac{(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac{B x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 110, normalized size = 1.3 \[{\frac{B{x}^{3}}{3\,{b}^{3}}}-{\frac{A{a}^{2}}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{B{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) Ba}{{b}^{4}}}+{\frac{2\,aA}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}B}{{b}^{4} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.37337, size = 127, normalized size = 1.44 \[ \frac{B x^{3}}{3 \, b^{3}} - \frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} - \frac{{\left (3 \, B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.223666, size = 192, normalized size = 2.18 \[ \frac{2 \, B b^{3} x^{9} + 4 \, B a b^{2} x^{6} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} - 2 \,{\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{6} + 3 \, B a^{3} - A a^{2} b + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.18242, size = 94, normalized size = 1.07 \[ \frac{B x^{3}}{3 b^{3}} - \frac{- 3 A a^{2} b + 5 B a^{3} + x^{3} \left (- 4 A a b^{2} + 6 B a^{2} b\right )}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} - \frac{\left (- A b + 3 B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218129, size = 126, normalized size = 1.43 \[ \frac{B x^{3}}{3 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} + \frac{9 \, B a b^{2} x^{6} - 3 \, A b^{3} x^{6} + 12 \, B a^{2} b x^{3} - 2 \, A a b^{2} x^{3} + 4 \, B a^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

1/3*B*x^3/b^3 - 1/3*(3*B*a - A*b)*ln(abs(b*x^3 + a))/b^4 + 1/6*(9*B*a*b^2*x^6 -
3*A*b^3*x^6 + 12*B*a^2*b*x^3 - 2*A*a*b^2*x^3 + 4*B*a^3)/((b*x^3 + a)^2*b^4)