3.268 \(\int \frac{\left (a+b x^3\right )^5}{x^{13}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0855881, antiderivative size = 66, 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 \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^13,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5}}{12 x^{12}} - \frac{5 a^{4} b}{9 x^{9}} - \frac{5 a^{3} b^{2}}{3 x^{6}} - \frac{10 a^{2} b^{3}}{3 x^{3}} + \frac{5 a b^{4} \log{\left (x^{3} \right )}}{3} + \frac{\int ^{x^{3}} b^{5}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5/x**13,x)

[Out]

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

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Mathematica [A]  time = 0.00889649, size = 66, normalized size = 1. \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^13,x]

[Out]

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

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Maple [A]  time = 0.01, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{5}}{12\,{x}^{12}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,{x}^{6}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}+{\frac{{b}^{5}{x}^{3}}{3}}+5\,a{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^13,x)

[Out]

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

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Maxima [A]  time = 1.43249, size = 82, normalized size = 1.24 \[ \frac{1}{3} \, b^{5} x^{3} + \frac{5}{3} \, a b^{4} \log \left (x^{3}\right ) - \frac{120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^13,x, algorithm="maxima")

[Out]

1/3*b^5*x^3 + 5/3*a*b^4*log(x^3) - 1/36*(120*a^2*b^3*x^9 + 60*a^3*b^2*x^6 + 20*a
^4*b*x^3 + 3*a^5)/x^12

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Fricas [A]  time = 0.214519, size = 82, normalized size = 1.24 \[ \frac{12 \, b^{5} x^{15} + 180 \, a b^{4} x^{12} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{9} - 60 \, a^{3} b^{2} x^{6} - 20 \, a^{4} b x^{3} - 3 \, a^{5}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^13,x, algorithm="fricas")

[Out]

1/36*(12*b^5*x^15 + 180*a*b^4*x^12*log(x) - 120*a^2*b^3*x^9 - 60*a^3*b^2*x^6 - 2
0*a^4*b*x^3 - 3*a^5)/x^12

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Sympy [A]  time = 2.18914, size = 61, normalized size = 0.92 \[ 5 a b^{4} \log{\left (x \right )} + \frac{b^{5} x^{3}}{3} - \frac{3 a^{5} + 20 a^{4} b x^{3} + 60 a^{3} b^{2} x^{6} + 120 a^{2} b^{3} x^{9}}{36 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**13,x)

[Out]

5*a*b**4*log(x) + b**5*x**3/3 - (3*a**5 + 20*a**4*b*x**3 + 60*a**3*b**2*x**6 + 1
20*a**2*b**3*x**9)/(36*x**12)

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GIAC/XCAS [A]  time = 0.222356, size = 93, normalized size = 1.41 \[ \frac{1}{3} \, b^{5} x^{3} + 5 \, a b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{125 \, a b^{4} x^{12} + 120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^13,x, algorithm="giac")

[Out]

1/3*b^5*x^3 + 5*a*b^4*ln(abs(x)) - 1/36*(125*a*b^4*x^12 + 120*a^2*b^3*x^9 + 60*a
^3*b^2*x^6 + 20*a^4*b*x^3 + 3*a^5)/x^12