3.59 \(\int \frac{\left (a+b x^2\right )^5}{x} \, dx\)

Optimal. Leaf size=65 \[ a^5 \log (x)+\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5
*x^10)/10 + a^5*Log[x]

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Rubi [A]  time = 0.0819982, 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 \[ a^5 \log (x)+\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x,x]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5
*x^10)/10 + a^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} \log{\left (x^{2} \right )}}{2} + \frac{5 a^{4} b x^{2}}{2} + 5 a^{3} b^{2} \int ^{x^{2}} x\, dx + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5/x,x)

[Out]

a**5*log(x**2)/2 + 5*a**4*b*x**2/2 + 5*a**3*b**2*Integral(x, (x, x**2)) + 5*a**2
*b**3*x**6/3 + 5*a*b**4*x**8/8 + b**5*x**10/10

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Mathematica [A]  time = 0.00721082, size = 65, normalized size = 1. \[ a^5 \log (x)+\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x,x]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5
*x^10)/10 + a^5*Log[x]

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \[{\frac{5\,{a}^{4}b{x}^{2}}{2}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{x}^{10}}{10}}+{a}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x,x)

[Out]

5/2*a^4*b*x^2+5/2*a^3*b^2*x^4+5/3*a^2*b^3*x^6+5/8*a*b^4*x^8+1/10*b^5*x^10+a^5*ln
(x)

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Maxima [A]  time = 1.33764, size = 78, normalized size = 1.2 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x,x, algorithm="maxima")

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^
2 + 1/2*a^5*log(x^2)

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Fricas [A]  time = 0.223893, size = 74, normalized size = 1.14 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x,x, algorithm="fricas")

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^
2 + a^5*log(x)

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Sympy [A]  time = 1.14663, size = 65, normalized size = 1. \[ a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{2}}{2} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x,x)

[Out]

a**5*log(x) + 5*a**4*b*x**2/2 + 5*a**3*b**2*x**4/2 + 5*a**2*b**3*x**6/3 + 5*a*b*
*4*x**8/8 + b**5*x**10/10

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GIAC/XCAS [A]  time = 0.207529, size = 78, normalized size = 1.2 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x,x, algorithm="giac")

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^
2 + 1/2*a^5*ln(x^2)