3.62 \(\int \frac{\left (a+b x^2\right )^5}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0899219, 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 \[ -\frac{a^5}{6 x^6}-\frac{5 a^4 b}{4 x^4}-\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00848019, size = 64, normalized size = 1. \[ -\frac{a^5}{6 x^6}-\frac{5 a^4 b}{4 x^4}-\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x^7,x)

[Out]

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

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Maxima [A]  time = 1.32544, size = 82, normalized size = 1.28 \[ \frac{1}{4} \, b^{5} x^{4} + \frac{5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac{60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221668, size = 82, normalized size = 1.28 \[ \frac{3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.67289, size = 63, normalized size = 0.98 \[ 10 a^{2} b^{3} \log{\left (x \right )} + \frac{5 a b^{4} x^{2}}{2} + \frac{b^{5} x^{4}}{4} - \frac{2 a^{5} + 15 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211455, size = 97, normalized size = 1.52 \[ \frac{1}{4} \, b^{5} x^{4} + \frac{5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3}{\rm ln}\left (x^{2}\right ) - \frac{110 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^7,x, algorithm="giac")

[Out]

1/4*b^5*x^4 + 5/2*a*b^4*x^2 + 5*a^2*b^3*ln(x^2) - 1/12*(110*a^2*b^3*x^6 + 60*a^3
*b^2*x^4 + 15*a^4*b*x^2 + 2*a^5)/x^6