3.300 \(\int \frac{\left (a+b x^3\right )^8}{x^{25}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a^8}{24 x^{24}}-\frac{8 a^7 b}{21 x^{21}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*
x^15) - (35*a^4*b^4)/(6*x^12) - (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8
*a*b^7)/(3*x^3) + b^8*Log[x]

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Rubi [A]  time = 0.116233, antiderivative size = 104, 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 \[ -\frac{a^8}{24 x^{24}}-\frac{8 a^7 b}{21 x^{21}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^25,x]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*
x^15) - (35*a^4*b^4)/(6*x^12) - (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8
*a*b^7)/(3*x^3) + b^8*Log[x]

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Rubi in Sympy [A]  time = 23.0771, size = 109, normalized size = 1.05 \[ - \frac{a^{8}}{24 x^{24}} - \frac{8 a^{7} b}{21 x^{21}} - \frac{14 a^{6} b^{2}}{9 x^{18}} - \frac{56 a^{5} b^{3}}{15 x^{15}} - \frac{35 a^{4} b^{4}}{6 x^{12}} - \frac{56 a^{3} b^{5}}{9 x^{9}} - \frac{14 a^{2} b^{6}}{3 x^{6}} - \frac{8 a b^{7}}{3 x^{3}} + \frac{b^{8} \log{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**25,x)

[Out]

-a**8/(24*x**24) - 8*a**7*b/(21*x**21) - 14*a**6*b**2/(9*x**18) - 56*a**5*b**3/(
15*x**15) - 35*a**4*b**4/(6*x**12) - 56*a**3*b**5/(9*x**9) - 14*a**2*b**6/(3*x**
6) - 8*a*b**7/(3*x**3) + b**8*log(x**3)/3

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Mathematica [A]  time = 0.00877809, size = 104, normalized size = 1. \[ -\frac{a^8}{24 x^{24}}-\frac{8 a^7 b}{21 x^{21}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^25,x]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*
x^15) - (35*a^4*b^4)/(6*x^12) - (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8
*a*b^7)/(3*x^3) + b^8*Log[x]

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Maple [A]  time = 0.011, size = 89, normalized size = 0.9 \[ -{\frac{{a}^{8}}{24\,{x}^{24}}}-{\frac{8\,{a}^{7}b}{21\,{x}^{21}}}-{\frac{14\,{a}^{6}{b}^{2}}{9\,{x}^{18}}}-{\frac{56\,{a}^{5}{b}^{3}}{15\,{x}^{15}}}-{\frac{35\,{a}^{4}{b}^{4}}{6\,{x}^{12}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{8\,a{b}^{7}}{3\,{x}^{3}}}+{b}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^25,x)

[Out]

-1/24*a^8/x^24-8/21*a^7*b/x^21-14/9*a^6*b^2/x^18-56/15*a^5*b^3/x^15-35/6*a^4*b^4
/x^12-56/9*a^3*b^5/x^9-14/3*a^2*b^6/x^6-8/3*a*b^7/x^3+b^8*ln(x)

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Maxima [A]  time = 1.44212, size = 127, normalized size = 1.22 \[ \frac{1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac{6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^25,x, algorithm="maxima")

[Out]

1/3*b^8*log(x^3) - 1/2520*(6720*a*b^7*x^21 + 11760*a^2*b^6*x^18 + 15680*a^3*b^5*
x^15 + 14700*a^4*b^4*x^12 + 9408*a^5*b^3*x^9 + 3920*a^6*b^2*x^6 + 960*a^7*b*x^3
+ 105*a^8)/x^24

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Fricas [A]  time = 0.210662, size = 127, normalized size = 1.22 \[ \frac{2520 \, b^{8} x^{24} \log \left (x\right ) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^25,x, algorithm="fricas")

[Out]

1/2520*(2520*b^8*x^24*log(x) - 6720*a*b^7*x^21 - 11760*a^2*b^6*x^18 - 15680*a^3*
b^5*x^15 - 14700*a^4*b^4*x^12 - 9408*a^5*b^3*x^9 - 3920*a^6*b^2*x^6 - 960*a^7*b*
x^3 - 105*a^8)/x^24

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Sympy [A]  time = 4.64219, size = 95, normalized size = 0.91 \[ b^{8} \log{\left (x \right )} - \frac{105 a^{8} + 960 a^{7} b x^{3} + 3920 a^{6} b^{2} x^{6} + 9408 a^{5} b^{3} x^{9} + 14700 a^{4} b^{4} x^{12} + 15680 a^{3} b^{5} x^{15} + 11760 a^{2} b^{6} x^{18} + 6720 a b^{7} x^{21}}{2520 x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**25,x)

[Out]

b**8*log(x) - (105*a**8 + 960*a**7*b*x**3 + 3920*a**6*b**2*x**6 + 9408*a**5*b**3
*x**9 + 14700*a**4*b**4*x**12 + 15680*a**3*b**5*x**15 + 11760*a**2*b**6*x**18 +
6720*a*b**7*x**21)/(2520*x**24)

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GIAC/XCAS [A]  time = 0.218652, size = 135, normalized size = 1.3 \[ b^{8}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^25,x, algorithm="giac")

[Out]

b^8*ln(abs(x)) - 1/2520*(2283*b^8*x^24 + 6720*a*b^7*x^21 + 11760*a^2*b^6*x^18 +
15680*a^3*b^5*x^15 + 14700*a^4*b^4*x^12 + 9408*a^5*b^3*x^9 + 3920*a^6*b^2*x^6 +
960*a^7*b*x^3 + 105*a^8)/x^24