3.42 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{10}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]

[Out]

-(a^5*A)/(9*x^9) - (a^4*(5*A*b + a*B))/(6*x^6) - (5*a^3*b*(2*A*b + a*B))/(3*x^3)
+ (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^9)/9 + 1
0*a^2*b^2*(A*b + a*B)*Log[x]

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Rubi [A]  time = 0.308313, antiderivative size = 114, 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 \[ -\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^10,x]

[Out]

-(a^5*A)/(9*x^9) - (a^4*(5*A*b + a*B))/(6*x^6) - (5*a^3*b*(2*A*b + a*B))/(3*x^3)
+ (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^9)/9 + 1
0*a^2*b^2*(A*b + a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{9 x^{9}} + \frac{B b^{5} x^{9}}{9} - \frac{a^{4} \left (5 A b + B a\right )}{6 x^{6}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{3 x^{3}} + \frac{10 a^{2} b^{2} \left (A b + B a\right ) \log{\left (x^{3} \right )}}{3} + \frac{5 a b^{3} x^{3} \left (A b + 2 B a\right )}{3} + \frac{b^{4} \left (A b + 5 B a\right ) \int ^{x^{3}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**10,x)

[Out]

-A*a**5/(9*x**9) + B*b**5*x**9/9 - a**4*(5*A*b + B*a)/(6*x**6) - 5*a**3*b*(2*A*b
+ B*a)/(3*x**3) + 10*a**2*b**2*(A*b + B*a)*log(x**3)/3 + 5*a*b**3*x**3*(A*b + 2
*B*a)/3 + b**4*(A*b + 5*B*a)*Integral(x, (x, x**3))/3

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Mathematica [A]  time = 0.120791, size = 106, normalized size = 0.93 \[ \frac{1}{18} \left (-\frac{2 a^5 A}{x^9}-\frac{3 a^4 (a B+5 A b)}{x^6}-\frac{30 a^3 b (a B+2 A b)}{x^3}+180 a^2 b^2 \log (x) (a B+A b)+3 b^4 x^6 (5 a B+A b)+30 a b^3 x^3 (2 a B+A b)+2 b^5 B x^9\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^10,x]

[Out]

((-2*a^5*A)/x^9 - (3*a^4*(5*A*b + a*B))/x^6 - (30*a^3*b*(2*A*b + a*B))/x^3 + 30*
a*b^3*(A*b + 2*a*B)*x^3 + 3*b^4*(A*b + 5*a*B)*x^6 + 2*b^5*B*x^9 + 180*a^2*b^2*(A
*b + a*B)*Log[x])/18

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Maple [A]  time = 0.014, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{9}}{9}}+{\frac{A{x}^{6}{b}^{5}}{6}}+{\frac{5\,B{x}^{6}a{b}^{4}}{6}}+{\frac{5\,A{x}^{3}a{b}^{4}}{3}}+{\frac{10\,B{x}^{3}{a}^{2}{b}^{3}}{3}}+10\,A\ln \left ( x \right ){a}^{2}{b}^{3}+10\,B\ln \left ( x \right ){a}^{3}{b}^{2}-{\frac{5\,{a}^{4}bA}{6\,{x}^{6}}}-{\frac{{a}^{5}B}{6\,{x}^{6}}}-{\frac{A{a}^{5}}{9\,{x}^{9}}}-{\frac{10\,{a}^{3}{b}^{2}A}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}bB}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^10,x)

[Out]

1/9*b^5*B*x^9+1/6*A*x^6*b^5+5/6*B*x^6*a*b^4+5/3*A*x^3*a*b^4+10/3*B*x^3*a^2*b^3+1
0*A*ln(x)*a^2*b^3+10*B*ln(x)*a^3*b^2-5/6*a^4/x^6*A*b-1/6*a^5/x^6*B-1/9*a^5*A/x^9
-10/3*a^3*b^2/x^3*A-5/3*a^4*b/x^3*B

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Maxima [A]  time = 1.40433, size = 166, normalized size = 1.46 \[ \frac{1}{9} \, B b^{5} x^{9} + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + \frac{5}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{3}\right ) - \frac{30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2 \, A a^{5} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{18 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^10,x, algorithm="maxima")

[Out]

1/9*B*b^5*x^9 + 1/6*(5*B*a*b^4 + A*b^5)*x^6 + 5/3*(2*B*a^2*b^3 + A*a*b^4)*x^3 +
10/3*(B*a^3*b^2 + A*a^2*b^3)*log(x^3) - 1/18*(30*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 2
*A*a^5 + 3*(B*a^5 + 5*A*a^4*b)*x^3)/x^9

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Fricas [A]  time = 0.222941, size = 166, normalized size = 1.46 \[ \frac{2 \, B b^{5} x^{18} + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 180 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} \log \left (x\right ) - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 2 \, A a^{5} - 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{18 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^10,x, algorithm="fricas")

[Out]

1/18*(2*B*b^5*x^18 + 3*(5*B*a*b^4 + A*b^5)*x^15 + 30*(2*B*a^2*b^3 + A*a*b^4)*x^1
2 + 180*(B*a^3*b^2 + A*a^2*b^3)*x^9*log(x) - 30*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 2*
A*a^5 - 3*(B*a^5 + 5*A*a^4*b)*x^3)/x^9

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Sympy [A]  time = 9.15999, size = 126, normalized size = 1.11 \[ \frac{B b^{5} x^{9}}{9} + 10 a^{2} b^{2} \left (A b + B a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A b^{5}}{6} + \frac{5 B a b^{4}}{6}\right ) + x^{3} \left (\frac{5 A a b^{4}}{3} + \frac{10 B a^{2} b^{3}}{3}\right ) - \frac{2 A a^{5} + x^{6} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x^{3} \left (15 A a^{4} b + 3 B a^{5}\right )}{18 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**10,x)

[Out]

B*b**5*x**9/9 + 10*a**2*b**2*(A*b + B*a)*log(x) + x**6*(A*b**5/6 + 5*B*a*b**4/6)
+ x**3*(5*A*a*b**4/3 + 10*B*a**2*b**3/3) - (2*A*a**5 + x**6*(60*A*a**3*b**2 + 3
0*B*a**4*b) + x**3*(15*A*a**4*b + 3*B*a**5))/(18*x**9)

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GIAC/XCAS [A]  time = 0.220218, size = 203, normalized size = 1.78 \[ \frac{1}{9} \, B b^{5} x^{9} + \frac{5}{6} \, B a b^{4} x^{6} + \frac{1}{6} \, A b^{5} x^{6} + \frac{10}{3} \, B a^{2} b^{3} x^{3} + \frac{5}{3} \, A a b^{4} x^{3} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{110 \, B a^{3} b^{2} x^{9} + 110 \, A a^{2} b^{3} x^{9} + 30 \, B a^{4} b x^{6} + 60 \, A a^{3} b^{2} x^{6} + 3 \, B a^{5} x^{3} + 15 \, A a^{4} b x^{3} + 2 \, A a^{5}}{18 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^10,x, algorithm="giac")

[Out]

1/9*B*b^5*x^9 + 5/6*B*a*b^4*x^6 + 1/6*A*b^5*x^6 + 10/3*B*a^2*b^3*x^3 + 5/3*A*a*b
^4*x^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*ln(abs(x)) - 1/18*(110*B*a^3*b^2*x^9 + 110*A
*a^2*b^3*x^9 + 30*B*a^4*b*x^6 + 60*A*a^3*b^2*x^6 + 3*B*a^5*x^3 + 15*A*a^4*b*x^3
+ 2*A*a^5)/x^9