3.97 \(\int \frac{\left (a+b x^2\right )^8}{x^{11}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^8}{10 x^{10}}-\frac{a^7 b}{x^8}-\frac{14 a^6 b^2}{3 x^6}-\frac{14 a^5 b^3}{x^4}-\frac{35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac{b^8 x^6}{6} \]

[Out]

-a^8/(10*x^10) - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4
*b^4)/x^2 + 14*a^2*b^6*x^2 + 2*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]

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Rubi [A]  time = 0.14399, antiderivative size = 95, 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 \[ -\frac{a^8}{10 x^{10}}-\frac{a^7 b}{x^8}-\frac{14 a^6 b^2}{3 x^6}-\frac{14 a^5 b^3}{x^4}-\frac{35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac{b^8 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^8/x^11,x]

[Out]

-a^8/(10*x^10) - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4
*b^4)/x^2 + 14*a^2*b^6*x^2 + 2*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{10 x^{10}} - \frac{a^{7} b}{x^{8}} - \frac{14 a^{6} b^{2}}{3 x^{6}} - \frac{14 a^{5} b^{3}}{x^{4}} - \frac{35 a^{4} b^{4}}{x^{2}} + 28 a^{3} b^{5} \log{\left (x^{2} \right )} + 14 a^{2} b^{6} x^{2} + 4 a b^{7} \int ^{x^{2}} x\, dx + \frac{b^{8} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**8/x**11,x)

[Out]

-a**8/(10*x**10) - a**7*b/x**8 - 14*a**6*b**2/(3*x**6) - 14*a**5*b**3/x**4 - 35*
a**4*b**4/x**2 + 28*a**3*b**5*log(x**2) + 14*a**2*b**6*x**2 + 4*a*b**7*Integral(
x, (x, x**2)) + b**8*x**6/6

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Mathematica [A]  time = 0.00847667, size = 95, normalized size = 1. \[ -\frac{a^8}{10 x^{10}}-\frac{a^7 b}{x^8}-\frac{14 a^6 b^2}{3 x^6}-\frac{14 a^5 b^3}{x^4}-\frac{35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac{b^8 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^8/x^11,x]

[Out]

-a^8/(10*x^10) - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4
*b^4)/x^2 + 14*a^2*b^6*x^2 + 2*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]

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Maple [A]  time = 0.011, size = 90, normalized size = 1. \[ -{\frac{{a}^{8}}{10\,{x}^{10}}}-{\frac{{a}^{7}b}{{x}^{8}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-14\,{\frac{{a}^{5}{b}^{3}}{{x}^{4}}}-35\,{\frac{{a}^{4}{b}^{4}}{{x}^{2}}}+14\,{a}^{2}{b}^{6}{x}^{2}+2\,a{b}^{7}{x}^{4}+{\frac{{b}^{8}{x}^{6}}{6}}+56\,{a}^{3}{b}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^8/x^11,x)

[Out]

-1/10*a^8/x^10-a^7*b/x^8-14/3*a^6*b^2/x^6-14*a^5*b^3/x^4-35*a^4*b^4/x^2+14*a^2*b
^6*x^2+2*a*b^7*x^4+1/6*b^8*x^6+56*a^3*b^5*ln(x)

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Maxima [A]  time = 1.3438, size = 127, normalized size = 1.34 \[ \frac{1}{6} \, b^{8} x^{6} + 2 \, a b^{7} x^{4} + 14 \, a^{2} b^{6} x^{2} + 28 \, a^{3} b^{5} \log \left (x^{2}\right ) - \frac{1050 \, a^{4} b^{4} x^{8} + 420 \, a^{5} b^{3} x^{6} + 140 \, a^{6} b^{2} x^{4} + 30 \, a^{7} b x^{2} + 3 \, a^{8}}{30 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^11,x, algorithm="maxima")

[Out]

1/6*b^8*x^6 + 2*a*b^7*x^4 + 14*a^2*b^6*x^2 + 28*a^3*b^5*log(x^2) - 1/30*(1050*a^
4*b^4*x^8 + 420*a^5*b^3*x^6 + 140*a^6*b^2*x^4 + 30*a^7*b*x^2 + 3*a^8)/x^10

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Fricas [A]  time = 0.202421, size = 127, normalized size = 1.34 \[ \frac{5 \, b^{8} x^{16} + 60 \, a b^{7} x^{14} + 420 \, a^{2} b^{6} x^{12} + 1680 \, a^{3} b^{5} x^{10} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{8} - 420 \, a^{5} b^{3} x^{6} - 140 \, a^{6} b^{2} x^{4} - 30 \, a^{7} b x^{2} - 3 \, a^{8}}{30 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^11,x, algorithm="fricas")

[Out]

1/30*(5*b^8*x^16 + 60*a*b^7*x^14 + 420*a^2*b^6*x^12 + 1680*a^3*b^5*x^10*log(x) -
1050*a^4*b^4*x^8 - 420*a^5*b^3*x^6 - 140*a^6*b^2*x^4 - 30*a^7*b*x^2 - 3*a^8)/x^
10

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Sympy [A]  time = 2.55012, size = 97, normalized size = 1.02 \[ 56 a^{3} b^{5} \log{\left (x \right )} + 14 a^{2} b^{6} x^{2} + 2 a b^{7} x^{4} + \frac{b^{8} x^{6}}{6} - \frac{3 a^{8} + 30 a^{7} b x^{2} + 140 a^{6} b^{2} x^{4} + 420 a^{5} b^{3} x^{6} + 1050 a^{4} b^{4} x^{8}}{30 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**8/x**11,x)

[Out]

56*a**3*b**5*log(x) + 14*a**2*b**6*x**2 + 2*a*b**7*x**4 + b**8*x**6/6 - (3*a**8
+ 30*a**7*b*x**2 + 140*a**6*b**2*x**4 + 420*a**5*b**3*x**6 + 1050*a**4*b**4*x**8
)/(30*x**10)

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GIAC/XCAS [A]  time = 0.207923, size = 142, normalized size = 1.49 \[ \frac{1}{6} \, b^{8} x^{6} + 2 \, a b^{7} x^{4} + 14 \, a^{2} b^{6} x^{2} + 28 \, a^{3} b^{5}{\rm ln}\left (x^{2}\right ) - \frac{1918 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 420 \, a^{5} b^{3} x^{6} + 140 \, a^{6} b^{2} x^{4} + 30 \, a^{7} b x^{2} + 3 \, a^{8}}{30 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^11,x, algorithm="giac")

[Out]

1/6*b^8*x^6 + 2*a*b^7*x^4 + 14*a^2*b^6*x^2 + 28*a^3*b^5*ln(x^2) - 1/30*(1918*a^3
*b^5*x^10 + 1050*a^4*b^4*x^8 + 420*a^5*b^3*x^6 + 140*a^6*b^2*x^4 + 30*a^7*b*x^2
+ 3*a^8)/x^10