3.95 \(\int \frac{\left (a+b x^2\right )^8}{x^7} \, dx\)

Optimal. Leaf size=94 \[ -\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.147566, antiderivative size = 94, 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 \[ -\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^8/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**8/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00846451, size = 94, normalized size = 1. \[ -\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^8/x^7,x]

[Out]

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

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Maple [A]  time = 0.01, size = 89, normalized size = 1. \[ -{\frac{{a}^{8}}{6\,{x}^{6}}}-2\,{\frac{{a}^{7}b}{{x}^{4}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}+35\,{a}^{4}{b}^{4}{x}^{2}+14\,{a}^{3}{b}^{5}{x}^{4}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+a{b}^{7}{x}^{8}+{\frac{{b}^{8}{x}^{10}}{10}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^8/x^7,x)

[Out]

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

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Maxima [A]  time = 1.3527, size = 123, normalized size = 1.31 \[ \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac{84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^7,x, algorithm="maxima")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 +
28*a^5*b^3*log(x^2) - 1/6*(84*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8)/x^6

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Fricas [A]  time = 0.201199, size = 127, normalized size = 1.35 \[ \frac{3 \, b^{8} x^{16} + 30 \, a b^{7} x^{14} + 140 \, a^{2} b^{6} x^{12} + 420 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{4} - 60 \, a^{7} b x^{2} - 5 \, a^{8}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.88127, size = 95, normalized size = 1.01 \[ 56 a^{5} b^{3} \log{\left (x \right )} + 35 a^{4} b^{4} x^{2} + 14 a^{3} b^{5} x^{4} + \frac{14 a^{2} b^{6} x^{6}}{3} + a b^{7} x^{8} + \frac{b^{8} x^{10}}{10} - \frac{a^{8} + 12 a^{7} b x^{2} + 84 a^{6} b^{2} x^{4}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**8/x**7,x)

[Out]

56*a**5*b**3*log(x) + 35*a**4*b**4*x**2 + 14*a**3*b**5*x**4 + 14*a**2*b**6*x**6/
3 + a*b**7*x**8 + b**8*x**10/10 - (a**8 + 12*a**7*b*x**2 + 84*a**6*b**2*x**4)/(6
*x**6)

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GIAC/XCAS [A]  time = 0.209781, size = 138, normalized size = 1.47 \[ \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3}{\rm ln}\left (x^{2}\right ) - \frac{308 \, a^{5} b^{3} x^{6} + 84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^7,x, algorithm="giac")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 +
28*a^5*b^3*ln(x^2) - 1/6*(308*a^5*b^3*x^6 + 84*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8
)/x^6