3.269 \(\int \frac{\left (a+b x^3\right )^5}{x^{16}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{12 x^{12}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a b^4}{3 x^3}+b^5 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0741138, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{12 x^{12}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a b^4}{3 x^3}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^16,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.5681, size = 68, normalized size = 1.05 \[ - \frac{a^{5}}{15 x^{15}} - \frac{5 a^{4} b}{12 x^{12}} - \frac{10 a^{3} b^{2}}{9 x^{9}} - \frac{5 a^{2} b^{3}}{3 x^{6}} - \frac{5 a b^{4}}{3 x^{3}} + \frac{b^{5} \log{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5/x**16,x)

[Out]

-a**5/(15*x**15) - 5*a**4*b/(12*x**12) - 10*a**3*b**2/(9*x**9) - 5*a**2*b**3/(3*
x**6) - 5*a*b**4/(3*x**3) + b**5*log(x**3)/3

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Mathematica [A]  time = 0.00807765, size = 65, normalized size = 1. \[ -\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{12 x^{12}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a b^4}{3 x^3}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^16,x]

[Out]

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

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Maple [A]  time = 0.01, size = 56, normalized size = 0.9 \[ -{\frac{{a}^{5}}{15\,{x}^{15}}}-{\frac{5\,{a}^{4}b}{12\,{x}^{12}}}-{\frac{10\,{a}^{3}{b}^{2}}{9\,{x}^{9}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,{x}^{6}}}-{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}+{b}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^16,x)

[Out]

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

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Maxima [A]  time = 1.43772, size = 82, normalized size = 1.26 \[ \frac{1}{3} \, b^{5} \log \left (x^{3}\right ) - \frac{300 \, a b^{4} x^{12} + 300 \, a^{2} b^{3} x^{9} + 200 \, a^{3} b^{2} x^{6} + 75 \, a^{4} b x^{3} + 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^16,x, algorithm="maxima")

[Out]

1/3*b^5*log(x^3) - 1/180*(300*a*b^4*x^12 + 300*a^2*b^3*x^9 + 200*a^3*b^2*x^6 + 7
5*a^4*b*x^3 + 12*a^5)/x^15

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Fricas [A]  time = 0.210876, size = 82, normalized size = 1.26 \[ \frac{180 \, b^{5} x^{15} \log \left (x\right ) - 300 \, a b^{4} x^{12} - 300 \, a^{2} b^{3} x^{9} - 200 \, a^{3} b^{2} x^{6} - 75 \, a^{4} b x^{3} - 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^16,x, algorithm="fricas")

[Out]

1/180*(180*b^5*x^15*log(x) - 300*a*b^4*x^12 - 300*a^2*b^3*x^9 - 200*a^3*b^2*x^6
- 75*a^4*b*x^3 - 12*a^5)/x^15

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Sympy [A]  time = 2.56291, size = 60, normalized size = 0.92 \[ b^{5} \log{\left (x \right )} - \frac{12 a^{5} + 75 a^{4} b x^{3} + 200 a^{3} b^{2} x^{6} + 300 a^{2} b^{3} x^{9} + 300 a b^{4} x^{12}}{180 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**16,x)

[Out]

b**5*log(x) - (12*a**5 + 75*a**4*b*x**3 + 200*a**3*b**2*x**6 + 300*a**2*b**3*x**
9 + 300*a*b**4*x**12)/(180*x**15)

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GIAC/XCAS [A]  time = 0.220102, size = 90, normalized size = 1.38 \[ b^{5}{\rm ln}\left ({\left | x \right |}\right ) - \frac{137 \, b^{5} x^{15} + 300 \, a b^{4} x^{12} + 300 \, a^{2} b^{3} x^{9} + 200 \, a^{3} b^{2} x^{6} + 75 \, a^{4} b x^{3} + 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^16,x, algorithm="giac")

[Out]

b^5*ln(abs(x)) - 1/180*(137*b^5*x^15 + 300*a*b^4*x^12 + 300*a^2*b^3*x^9 + 200*a^
3*b^2*x^6 + 75*a^4*b*x^3 + 12*a^5)/x^15