3.60 \(\int \frac{\left (a+b x^2\right )^5}{x^3} \, dx\)

Optimal. Leaf size=64 \[ -\frac{a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]

[Out]

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

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Rubi [A]  time = 0.0925029, 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 \[ -\frac{a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.00822996, size = 64, normalized size = 1. \[ -\frac{a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{5}}{2\,{x}^{2}}}+5\,{a}^{3}{b}^{2}{x}^{2}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{{b}^{5}{x}^{8}}{8}}+5\,{a}^{4}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x^3,x)

[Out]

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

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Maxima [A]  time = 1.33178, size = 78, normalized size = 1.22 \[ \frac{1}{8} \, b^{5} x^{8} + \frac{5}{6} \, a b^{4} x^{6} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac{5}{2} \, a^{4} b \log \left (x^{2}\right ) - \frac{a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.217206, size = 82, normalized size = 1.28 \[ \frac{3 \, b^{5} x^{10} + 20 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} \log \left (x\right ) - 12 \, a^{5}}{24 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^3,x, algorithm="fricas")

[Out]

1/24*(3*b^5*x^10 + 20*a*b^4*x^8 + 60*a^2*b^3*x^6 + 120*a^3*b^2*x^4 + 120*a^4*b*x
^2*log(x) - 12*a^5)/x^2

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Sympy [A]  time = 1.25594, size = 63, normalized size = 0.98 \[ - \frac{a^{5}}{2 x^{2}} + 5 a^{4} b \log{\left (x \right )} + 5 a^{3} b^{2} x^{2} + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213439, size = 92, normalized size = 1.44 \[ \frac{1}{8} \, b^{5} x^{8} + \frac{5}{6} \, a b^{4} x^{6} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac{5}{2} \, a^{4} b{\rm ln}\left (x^{2}\right ) - \frac{5 \, a^{4} b x^{2} + a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^3,x, algorithm="giac")

[Out]

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