3.88 \(\int \frac{x^{11} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^6}{6 b^3} \]

[Out]

((A*b - 3*a*B)*x^3)/(3*b^4) + (B*x^6)/(6*b^3) + (a^3*(A*b - a*B))/(6*b^5*(a + b*
x^3)^2) - (a^2*(3*A*b - 4*a*B))/(3*b^5*(a + b*x^3)) - (a*(A*b - 2*a*B)*Log[a + b
*x^3])/b^5

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Rubi [A]  time = 0.346, antiderivative size = 107, 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 \[ \frac{a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac{a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

((A*b - 3*a*B)*x^3)/(3*b^4) + (B*x^6)/(6*b^3) + (a^3*(A*b - a*B))/(6*b^5*(a + b*
x^3)^2) - (a^2*(3*A*b - 4*a*B))/(3*b^5*(a + b*x^3)) - (a*(A*b - 2*a*B)*Log[a + b
*x^3])/b^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{3}} x\, dx}{3 b^{3}} + \frac{a^{3} \left (A b - B a\right )}{6 b^{5} \left (a + b x^{3}\right )^{2}} - \frac{a^{2} \left (3 A b - 4 B a\right )}{3 b^{5} \left (a + b x^{3}\right )} - \frac{a \left (A b - 2 B a\right ) \log{\left (a + b x^{3} \right )}}{b^{5}} + \left (\frac{A b}{3} - B a\right ) \int ^{x^{3}} \frac{1}{b^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.131595, size = 94, normalized size = 0.88 \[ \frac{\frac{a^3 (A b-a B)}{\left (a+b x^3\right )^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x^3}+2 b x^3 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(2*b*(A*b - 3*a*B)*x^3 + b^2*B*x^6 + (a^3*(A*b - a*B))/(a + b*x^3)^2 + (2*a^2*(-
3*A*b + 4*a*B))/(a + b*x^3) + 6*a*(-(A*b) + 2*a*B)*Log[a + b*x^3])/(6*b^5)

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Maple [A]  time = 0.012, size = 134, normalized size = 1.3 \[{\frac{B{x}^{6}}{6\,{b}^{3}}}+{\frac{A{x}^{3}}{3\,{b}^{3}}}-{\frac{B{x}^{3}a}{{b}^{4}}}+{\frac{A{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{B{a}^{4}}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) A}{{b}^{4}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{{b}^{5}}}-{\frac{A{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,B{a}^{3}}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.41061, size = 155, normalized size = 1.45 \[ \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{B b x^{6} - 2 \,{\left (3 \, B a - A b\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

1/6*(7*B*a^4 - 5*A*a^3*b + 2*(4*B*a^3*b - 3*A*a^2*b^2)*x^3)/(b^7*x^6 + 2*a*b^6*x
^3 + a^2*b^5) + 1/6*(B*b*x^6 - 2*(3*B*a - A*b)*x^3)/b^4 + (2*B*a^2 - A*a*b)*log(
b*x^3 + a)/b^5

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Fricas [A]  time = 0.220749, size = 242, normalized size = 2.26 \[ \frac{B b^{4} x^{12} - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{9} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 6 \,{\left ({\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} + 2 \, B a^{4} - A a^{3} b + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/6*(B*b^4*x^12 - 2*(2*B*a*b^3 - A*b^4)*x^9 - (11*B*a^2*b^2 - 4*A*a*b^3)*x^6 + 7
*B*a^4 - 5*A*a^3*b + 2*(B*a^3*b - 2*A*a^2*b^2)*x^3 + 6*((2*B*a^2*b^2 - A*a*b^3)*
x^6 + 2*B*a^4 - A*a^3*b + 2*(2*B*a^3*b - A*a^2*b^2)*x^3)*log(b*x^3 + a))/(b^7*x^
6 + 2*a*b^6*x^3 + a^2*b^5)

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Sympy [A]  time = 10.6876, size = 112, normalized size = 1.05 \[ \frac{B x^{6}}{6 b^{3}} + \frac{a \left (- A b + 2 B a\right ) \log{\left (a + b x^{3} \right )}}{b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x^{3} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{6 a^{2} b^{5} + 12 a b^{6} x^{3} + 6 b^{7} x^{6}} - \frac{x^{3} \left (- A b + 3 B a\right )}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**6/(6*b**3) + a*(-A*b + 2*B*a)*log(a + b*x**3)/b**5 + (-5*A*a**3*b + 7*B*a**
4 + x**3*(-6*A*a**2*b**2 + 8*B*a**3*b))/(6*a**2*b**5 + 12*a*b**6*x**3 + 6*b**7*x
**6) - x**3*(-A*b + 3*B*a)/(3*b**4)

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GIAC/XCAS [A]  time = 0.221361, size = 177, normalized size = 1.65 \[ \frac{{\left (2 \, B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{6} - 6 \, B a b^{2} x^{3} + 2 \, A b^{3} x^{3}}{6 \, b^{6}} - \frac{18 \, B a^{2} b^{2} x^{6} - 9 \, A a b^{3} x^{6} + 28 \, B a^{3} b x^{3} - 12 \, A a^{2} b^{2} x^{3} + 11 \, B a^{4} - 4 \, A a^{3} b}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^11/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

(2*B*a^2 - A*a*b)*ln(abs(b*x^3 + a))/b^5 + 1/6*(B*b^3*x^6 - 6*B*a*b^2*x^3 + 2*A*
b^3*x^3)/b^6 - 1/6*(18*B*a^2*b^2*x^6 - 9*A*a*b^3*x^6 + 28*B*a^3*b*x^3 - 12*A*a^2
*b^2*x^3 + 11*B*a^4 - 4*A*a^3*b)/((b*x^3 + a)^2*b^5)