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

Optimal. Leaf size=64 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.090487, antiderivative size = 64, 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 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0118605, size = 64, normalized size = 1. \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x^5,x]

[Out]

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

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Maple [A]  time = 0.009, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}+5\,{a}^{2}{b}^{3}{x}^{2}+{\frac{5\,a{b}^{4}{x}^{4}}{4}}+{\frac{{b}^{5}{x}^{6}}{6}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x^5,x)

[Out]

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

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Maxima [A]  time = 1.33321, size = 80, normalized size = 1.25 \[ \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac{10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22133, size = 82, normalized size = 1.28 \[ \frac{2 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} \log \left (x\right ) - 30 \, a^{4} b x^{2} - 3 \, a^{5}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^5*x^10 + 15*a*b^4*x^8 + 60*a^2*b^3*x^6 + 120*a^3*b^2*x^4*log(x) - 30*a
^4*b*x^2 - 3*a^5)/x^4

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Sympy [A]  time = 1.46812, size = 61, normalized size = 0.95 \[ 10 a^{3} b^{2} \log{\left (x \right )} + 5 a^{2} b^{3} x^{2} + \frac{5 a b^{4} x^{4}}{4} + \frac{b^{5} x^{6}}{6} - \frac{a^{5} + 10 a^{4} b x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209507, size = 95, normalized size = 1.48 \[ \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2}{\rm ln}\left (x^{2}\right ) - \frac{30 \, a^{3} b^{2} x^{4} + 10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^5*x^6 + 5/4*a*b^4*x^4 + 5*a^2*b^3*x^2 + 5*a^3*b^2*ln(x^2) - 1/4*(30*a^3*b^
2*x^4 + 10*a^4*b*x^2 + a^5)/x^4