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

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]

-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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Rubi [A]  time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a^7 b}{21 x^{21}}-\frac {a^8}{24 x^{24}}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \]

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^9} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^8}{x^9}+\frac {8 a^7 b}{x^8}+\frac {28 a^6 b^2}{x^7}+\frac {56 a^5 b^3}{x^6}+\frac {70 a^4 b^4}{x^5}+\frac {56 a^3 b^5}{x^4}+\frac {28 a^2 b^6}{x^3}+\frac {8 a b^7}{x^2}+\frac {b^8}{x}\right ) \, dx,x,x^3\right )\\ &=-\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)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 104, normalized size = 1.00 \[ -\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^25,x]

[Out]

-1/24*a^8/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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fricas [A]  time = 0.54, size = 94, normalized size = 0.90 \[ \frac {2520 \, b^{8} x^{24} \log \relax (x) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \]

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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giac [A]  time = 0.15, size = 100, normalized size = 0.96 \[ b^{8} \log \left ({\left | x \right |}\right ) - \frac {2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \]

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*log(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

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maple [A]  time = 0.01, size = 89, normalized size = 0.86 \[ b^{8} \ln \relax (x )-\frac {8 a \,b^{7}}{3 x^{3}}-\frac {14 a^{2} b^{6}}{3 x^{6}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {35 a^{4} b^{4}}{6 x^{12}}-\frac {56 a^{5} b^{3}}{15 x^{15}}-\frac {14 a^{6} b^{2}}{9 x^{18}}-\frac {8 a^{7} b}{21 x^{21}}-\frac {a^{8}}{24 x^{24}} \]

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 = 1.33, size = 94, normalized size = 0.90 \[ \frac {1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac {6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \]

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 + 94
08*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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mupad [B]  time = 0.07, size = 91, normalized size = 0.88 \[ b^8\,\ln \relax (x)-\frac {\frac {a^8}{24}+\frac {8\,a^7\,b\,x^3}{21}+\frac {14\,a^6\,b^2\,x^6}{9}+\frac {56\,a^5\,b^3\,x^9}{15}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{15}}{9}+\frac {14\,a^2\,b^6\,x^{18}}{3}+\frac {8\,a\,b^7\,x^{21}}{3}}{x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^25,x)

[Out]

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

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sympy [A]  time = 1.20, size = 97, normalized size = 0.93 \[ b^{8} \log {\relax (x )} + \frac {- 105 a^{8} - 960 a^{7} b x^{3} - 3920 a^{6} b^{2} x^{6} - 9408 a^{5} b^{3} x^{9} - 14700 a^{4} b^{4} x^{12} - 15680 a^{3} b^{5} x^{15} - 11760 a^{2} b^{6} x^{18} - 6720 a b^{7} x^{21}}{2520 x^{24}} \]

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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