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

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

[Out]

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

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Rubi [A]  time = 0.0751672, 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 \[ a^5 \log (x)+\frac{5}{3} a^4 b x^3+\frac{5}{3} a^3 b^2 x^6+\frac{10}{9} a^2 b^3 x^9+\frac{5}{12} a b^4 x^{12}+\frac{b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} \log{\left (x^{3} \right )}}{3} + \frac{5 a^{4} b x^{3}}{3} + \frac{10 a^{3} b^{2} \int ^{x^{3}} x\, dx}{3} + \frac{10 a^{2} b^{3} x^{9}}{9} + \frac{5 a b^{4} x^{12}}{12} + \frac{b^{5} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00738425, size = 65, normalized size = 1. \[ a^5 \log (x)+\frac{5}{3} a^4 b x^3+\frac{5}{3} a^3 b^2 x^6+\frac{10}{9} a^2 b^3 x^9+\frac{5}{12} a b^4 x^{12}+\frac{b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x,x]

[Out]

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

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Maple [A]  time = 0.003, size = 56, normalized size = 0.9 \[{\frac{5\,{a}^{4}b{x}^{3}}{3}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{6}}{3}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{9}}{9}}+{\frac{5\,a{b}^{4}{x}^{12}}{12}}+{\frac{{b}^{5}{x}^{15}}{15}}+{a}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x,x)

[Out]

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

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Maxima [A]  time = 1.42712, size = 78, normalized size = 1.2 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{3} \, a^{5} \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216725, size = 74, normalized size = 1.14 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.1515, size = 65, normalized size = 1. \[ a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{3}}{3} + \frac{5 a^{3} b^{2} x^{6}}{3} + \frac{10 a^{2} b^{3} x^{9}}{9} + \frac{5 a b^{4} x^{12}}{12} + \frac{b^{5} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223728, size = 76, normalized size = 1.17 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{12} \, a b^{4} x^{12} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{3} \, a^{3} b^{2} x^{6} + \frac{5}{3} \, a^{4} b x^{3} + a^{5}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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