3.108 \(\int \frac{\left (a+b x^2\right )^8}{x^{33}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) -
(35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b
^7)/(9*x^18) - b^8/(16*x^16)

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Rubi [A]  time = 0.136572, antiderivative size = 106, 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 \[ -\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^8/x^33,x]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) -
(35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b
^7)/(9*x^18) - b^8/(16*x^16)

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Rubi in Sympy [A]  time = 24.4739, size = 105, normalized size = 0.99 \[ - \frac{a^{8}}{32 x^{32}} - \frac{4 a^{7} b}{15 x^{30}} - \frac{a^{6} b^{2}}{x^{28}} - \frac{28 a^{5} b^{3}}{13 x^{26}} - \frac{35 a^{4} b^{4}}{12 x^{24}} - \frac{28 a^{3} b^{5}}{11 x^{22}} - \frac{7 a^{2} b^{6}}{5 x^{20}} - \frac{4 a b^{7}}{9 x^{18}} - \frac{b^{8}}{16 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**8/x**33,x)

[Out]

-a**8/(32*x**32) - 4*a**7*b/(15*x**30) - a**6*b**2/x**28 - 28*a**5*b**3/(13*x**2
6) - 35*a**4*b**4/(12*x**24) - 28*a**3*b**5/(11*x**22) - 7*a**2*b**6/(5*x**20) -
4*a*b**7/(9*x**18) - b**8/(16*x**16)

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Mathematica [A]  time = 0.00780919, size = 106, normalized size = 1. \[ -\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^8/x^33,x]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) -
(35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b
^7)/(9*x^18) - b^8/(16*x^16)

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Maple [A]  time = 0.01, size = 91, normalized size = 0.9 \[ -{\frac{{a}^{8}}{32\,{x}^{32}}}-{\frac{4\,{a}^{7}b}{15\,{x}^{30}}}-{\frac{{a}^{6}{b}^{2}}{{x}^{28}}}-{\frac{28\,{a}^{5}{b}^{3}}{13\,{x}^{26}}}-{\frac{35\,{a}^{4}{b}^{4}}{12\,{x}^{24}}}-{\frac{28\,{a}^{3}{b}^{5}}{11\,{x}^{22}}}-{\frac{7\,{a}^{2}{b}^{6}}{5\,{x}^{20}}}-{\frac{4\,a{b}^{7}}{9\,{x}^{18}}}-{\frac{{b}^{8}}{16\,{x}^{16}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^8/x^33,x)

[Out]

-1/32*a^8/x^32-4/15*a^7*b/x^30-a^6*b^2/x^28-28/13*a^5*b^3/x^26-35/12*a^4*b^4/x^2
4-28/11*a^3*b^5/x^22-7/5*a^2*b^6/x^20-4/9*a*b^7/x^18-1/16*b^8/x^16

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Maxima [A]  time = 1.34485, size = 124, normalized size = 1.17 \[ -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^33,x, algorithm="maxima")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*
b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*
a^7*b*x^2 + 6435*a^8)/x^32

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Fricas [A]  time = 0.19363, size = 124, normalized size = 1.17 \[ -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^33,x, algorithm="fricas")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*
b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*
a^7*b*x^2 + 6435*a^8)/x^32

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Sympy [A]  time = 5.85024, size = 99, normalized size = 0.93 \[ - \frac{6435 a^{8} + 54912 a^{7} b x^{2} + 205920 a^{6} b^{2} x^{4} + 443520 a^{5} b^{3} x^{6} + 600600 a^{4} b^{4} x^{8} + 524160 a^{3} b^{5} x^{10} + 288288 a^{2} b^{6} x^{12} + 91520 a b^{7} x^{14} + 12870 b^{8} x^{16}}{205920 x^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**8/x**33,x)

[Out]

-(6435*a**8 + 54912*a**7*b*x**2 + 205920*a**6*b**2*x**4 + 443520*a**5*b**3*x**6
+ 600600*a**4*b**4*x**8 + 524160*a**3*b**5*x**10 + 288288*a**2*b**6*x**12 + 9152
0*a*b**7*x**14 + 12870*b**8*x**16)/(205920*x**32)

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GIAC/XCAS [A]  time = 0.208936, size = 124, normalized size = 1.17 \[ -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^33,x, algorithm="giac")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*
b^5*x^10 + 600600*a^4*b^4*x^8 + 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*
a^7*b*x^2 + 6435*a^8)/x^32