3.51 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{a^5 B}{15 x^{15}}-\frac{5 a^4 b B}{12 x^{12}}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^2 b^3 B}{3 x^6}-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}-\frac{5 a b^4 B}{3 x^3}+b^5 B \log (x) \]

[Out]

-(a^5*B)/(15*x^15) - (5*a^4*b*B)/(12*x^12) - (10*a^3*b^2*B)/(9*x^9) - (5*a^2*b^3
*B)/(3*x^6) - (5*a*b^4*B)/(3*x^3) - (A*(a + b*x^3)^6)/(18*a*x^18) + b^5*B*Log[x]

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Rubi [A]  time = 0.156191, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{a^5 B}{15 x^{15}}-\frac{5 a^4 b B}{12 x^{12}}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^2 b^3 B}{3 x^6}-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}-\frac{5 a b^4 B}{3 x^3}+b^5 B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^19,x]

[Out]

-(a^5*B)/(15*x^15) - (5*a^4*b*B)/(12*x^12) - (10*a^3*b^2*B)/(9*x^9) - (5*a^2*b^3
*B)/(3*x^6) - (5*a*b^4*B)/(3*x^3) - (A*(a + b*x^3)^6)/(18*a*x^18) + b^5*B*Log[x]

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Rubi in Sympy [A]  time = 20.1783, size = 95, normalized size = 1.04 \[ - \frac{A \left (a + b x^{3}\right )^{6}}{18 a x^{18}} - \frac{B a^{5}}{15 x^{15}} - \frac{5 B a^{4} b}{12 x^{12}} - \frac{10 B a^{3} b^{2}}{9 x^{9}} - \frac{5 B a^{2} b^{3}}{3 x^{6}} - \frac{5 B a b^{4}}{3 x^{3}} + \frac{B b^{5} \log{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**19,x)

[Out]

-A*(a + b*x**3)**6/(18*a*x**18) - B*a**5/(15*x**15) - 5*B*a**4*b/(12*x**12) - 10
*B*a**3*b**2/(9*x**9) - 5*B*a**2*b**3/(3*x**6) - 5*B*a*b**4/(3*x**3) + B*b**5*lo
g(x**3)/3

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Mathematica [A]  time = 0.079167, size = 121, normalized size = 1.33 \[ -\frac{2 a^5 \left (5 A+6 B x^3\right )+15 a^4 b x^3 \left (4 A+5 B x^3\right )+50 a^3 b^2 x^6 \left (3 A+4 B x^3\right )+100 a^2 b^3 x^9 \left (2 A+3 B x^3\right )+150 a b^4 x^{12} \left (A+2 B x^3\right )+60 A b^5 x^{15}-180 b^5 B x^{18} \log (x)}{180 x^{18}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^19,x]

[Out]

-(60*A*b^5*x^15 + 150*a*b^4*x^12*(A + 2*B*x^3) + 100*a^2*b^3*x^9*(2*A + 3*B*x^3)
+ 50*a^3*b^2*x^6*(3*A + 4*B*x^3) + 15*a^4*b*x^3*(4*A + 5*B*x^3) + 2*a^5*(5*A +
6*B*x^3) - 180*b^5*B*x^18*Log[x])/(180*x^18)

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Maple [A]  time = 0.012, size = 124, normalized size = 1.4 \[ -{\frac{5\,{a}^{3}{b}^{2}A}{6\,{x}^{12}}}-{\frac{5\,{a}^{4}bB}{12\,{x}^{12}}}-{\frac{{a}^{4}bA}{3\,{x}^{15}}}-{\frac{{a}^{5}B}{15\,{x}^{15}}}+{b}^{5}B\ln \left ( x \right ) -{\frac{5\,a{b}^{4}A}{6\,{x}^{6}}}-{\frac{5\,{a}^{2}{b}^{3}B}{3\,{x}^{6}}}-{\frac{10\,{a}^{2}{b}^{3}A}{9\,{x}^{9}}}-{\frac{10\,{a}^{3}{b}^{2}B}{9\,{x}^{9}}}-{\frac{{b}^{5}A}{3\,{x}^{3}}}-{\frac{5\,a{b}^{4}B}{3\,{x}^{3}}}-{\frac{A{a}^{5}}{18\,{x}^{18}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^19,x)

[Out]

-5/6*a^3*b^2/x^12*A-5/12*a^4*b*B/x^12-1/3*a^4/x^15*A*b-1/15*a^5*B/x^15+b^5*B*ln(
x)-5/6*a*b^4/x^6*A-5/3*a^2*b^3*B/x^6-10/9*a^2*b^3/x^9*A-10/9*a^3*b^2*B/x^9-1/3*b
^5/x^3*A-5/3*a*b^4*B/x^3-1/18*A*a^5/x^18

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Maxima [A]  time = 1.43841, size = 166, normalized size = 1.82 \[ \frac{1}{3} \, B b^{5} \log \left (x^{3}\right ) - \frac{60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 10 \, A a^{5} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^19,x, algorithm="maxima")

[Out]

1/3*B*b^5*log(x^3) - 1/180*(60*(5*B*a*b^4 + A*b^5)*x^15 + 150*(2*B*a^2*b^3 + A*a
*b^4)*x^12 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^6 +
10*A*a^5 + 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18

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Fricas [A]  time = 0.225584, size = 166, normalized size = 1.82 \[ \frac{180 \, B b^{5} x^{18} \log \left (x\right ) - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 10 \, A a^{5} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^19,x, algorithm="fricas")

[Out]

1/180*(180*B*b^5*x^18*log(x) - 60*(5*B*a*b^4 + A*b^5)*x^15 - 150*(2*B*a^2*b^3 +
A*a*b^4)*x^12 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^6
- 10*A*a^5 - 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**19,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214539, size = 184, normalized size = 2.02 \[ B b^{5}{\rm ln}\left ({\left | x \right |}\right ) - \frac{147 \, B b^{5} x^{18} + 300 \, B a b^{4} x^{15} + 60 \, A b^{5} x^{15} + 300 \, B a^{2} b^{3} x^{12} + 150 \, A a b^{4} x^{12} + 200 \, B a^{3} b^{2} x^{9} + 200 \, A a^{2} b^{3} x^{9} + 75 \, B a^{4} b x^{6} + 150 \, A a^{3} b^{2} x^{6} + 12 \, B a^{5} x^{3} + 60 \, A a^{4} b x^{3} + 10 \, A a^{5}}{180 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^19,x, algorithm="giac")

[Out]

B*b^5*ln(abs(x)) - 1/180*(147*B*b^5*x^18 + 300*B*a*b^4*x^15 + 60*A*b^5*x^15 + 30
0*B*a^2*b^3*x^12 + 150*A*a*b^4*x^12 + 200*B*a^3*b^2*x^9 + 200*A*a^2*b^3*x^9 + 75
*B*a^4*b*x^6 + 150*A*a^3*b^2*x^6 + 12*B*a^5*x^3 + 60*A*a^4*b*x^3 + 10*A*a^5)/x^1
8