3.307 \(\int \frac {(a+b x^3)^8}{x^{46}} \, dx\)

Optimal. Leaf size=108 \[ -\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \]

[Out]

-1/45*a^8/x^45-4/21*a^7*b/x^42-28/39*a^6*b^2/x^39-14/9*a^5*b^3/x^36-70/33*a^4*b^4/x^33-28/15*a^3*b^5/x^30-28/2
7*a^2*b^6/x^27-1/3*a*b^7/x^24-1/21*b^8/x^21

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Rubi [A]  time = 0.05, antiderivative size = 108, 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} \[ -\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {4 a^7 b}{21 x^{42}}-\frac {a^8}{45 x^{45}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^46,x]

[Out]

-a^8/(45*x^45) - (4*a^7*b)/(21*x^42) - (28*a^6*b^2)/(39*x^39) - (14*a^5*b^3)/(9*x^36) - (70*a^4*b^4)/(33*x^33)
- (28*a^3*b^5)/(15*x^30) - (28*a^2*b^6)/(27*x^27) - (a*b^7)/(3*x^24) - b^8/(21*x^21)

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^{46}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{16}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^8}{x^{16}}+\frac {8 a^7 b}{x^{15}}+\frac {28 a^6 b^2}{x^{14}}+\frac {56 a^5 b^3}{x^{13}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^3 b^5}{x^{11}}+\frac {28 a^2 b^6}{x^{10}}+\frac {8 a b^7}{x^9}+\frac {b^8}{x^8}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 108, normalized size = 1.00 \[ -\frac {a^8}{45 x^{45}}-\frac {4 a^7 b}{21 x^{42}}-\frac {28 a^6 b^2}{39 x^{39}}-\frac {14 a^5 b^3}{9 x^{36}}-\frac {70 a^4 b^4}{33 x^{33}}-\frac {28 a^3 b^5}{15 x^{30}}-\frac {28 a^2 b^6}{27 x^{27}}-\frac {a b^7}{3 x^{24}}-\frac {b^8}{21 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^46,x]

[Out]

-1/45*a^8/x^45 - (4*a^7*b)/(21*x^42) - (28*a^6*b^2)/(39*x^39) - (14*a^5*b^3)/(9*x^36) - (70*a^4*b^4)/(33*x^33)
- (28*a^3*b^5)/(15*x^30) - (28*a^2*b^6)/(27*x^27) - (a*b^7)/(3*x^24) - b^8/(21*x^21)

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fricas [A]  time = 0.59, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^46,x, algorithm="fricas")

[Out]

-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252*a^3*b^5*x^15 + 286650*a^4*b^4*x^12
+ 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45

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giac [A]  time = 0.19, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^46,x, algorithm="giac")

[Out]

-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252*a^3*b^5*x^15 + 286650*a^4*b^4*x^12
+ 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45

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maple [A]  time = 0.01, size = 91, normalized size = 0.84 \[ -\frac {b^{8}}{21 x^{21}}-\frac {a \,b^{7}}{3 x^{24}}-\frac {28 a^{2} b^{6}}{27 x^{27}}-\frac {28 a^{3} b^{5}}{15 x^{30}}-\frac {70 a^{4} b^{4}}{33 x^{33}}-\frac {14 a^{5} b^{3}}{9 x^{36}}-\frac {28 a^{6} b^{2}}{39 x^{39}}-\frac {4 a^{7} b}{21 x^{42}}-\frac {a^{8}}{45 x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^46,x)

[Out]

-1/45*a^8/x^45-4/21*a^7*b/x^42-28/39*a^6*b^2/x^39-14/9*a^5*b^3/x^36-70/33*a^4*b^4/x^33-28/15*a^3*b^5/x^30-28/2
7*a^2*b^6/x^27-1/3*a*b^7/x^24-1/21*b^8/x^21

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maxima [A]  time = 1.29, size = 92, normalized size = 0.85 \[ -\frac {6435 \, b^{8} x^{24} + 45045 \, a b^{7} x^{21} + 140140 \, a^{2} b^{6} x^{18} + 252252 \, a^{3} b^{5} x^{15} + 286650 \, a^{4} b^{4} x^{12} + 210210 \, a^{5} b^{3} x^{9} + 97020 \, a^{6} b^{2} x^{6} + 25740 \, a^{7} b x^{3} + 3003 \, a^{8}}{135135 \, x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^46,x, algorithm="maxima")

[Out]

-1/135135*(6435*b^8*x^24 + 45045*a*b^7*x^21 + 140140*a^2*b^6*x^18 + 252252*a^3*b^5*x^15 + 286650*a^4*b^4*x^12
+ 210210*a^5*b^3*x^9 + 97020*a^6*b^2*x^6 + 25740*a^7*b*x^3 + 3003*a^8)/x^45

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mupad [B]  time = 1.00, size = 92, normalized size = 0.85 \[ -\frac {\frac {a^8}{45}+\frac {4\,a^7\,b\,x^3}{21}+\frac {28\,a^6\,b^2\,x^6}{39}+\frac {14\,a^5\,b^3\,x^9}{9}+\frac {70\,a^4\,b^4\,x^{12}}{33}+\frac {28\,a^3\,b^5\,x^{15}}{15}+\frac {28\,a^2\,b^6\,x^{18}}{27}+\frac {a\,b^7\,x^{21}}{3}+\frac {b^8\,x^{24}}{21}}{x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^46,x)

[Out]

-(a^8/45 + (b^8*x^24)/21 + (4*a^7*b*x^3)/21 + (a*b^7*x^21)/3 + (28*a^6*b^2*x^6)/39 + (14*a^5*b^3*x^9)/9 + (70*
a^4*b^4*x^12)/33 + (28*a^3*b^5*x^15)/15 + (28*a^2*b^6*x^18)/27)/x^45

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sympy [A]  time = 1.92, size = 99, normalized size = 0.92 \[ \frac {- 3003 a^{8} - 25740 a^{7} b x^{3} - 97020 a^{6} b^{2} x^{6} - 210210 a^{5} b^{3} x^{9} - 286650 a^{4} b^{4} x^{12} - 252252 a^{3} b^{5} x^{15} - 140140 a^{2} b^{6} x^{18} - 45045 a b^{7} x^{21} - 6435 b^{8} x^{24}}{135135 x^{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**46,x)

[Out]

(-3003*a**8 - 25740*a**7*b*x**3 - 97020*a**6*b**2*x**6 - 210210*a**5*b**3*x**9 - 286650*a**4*b**4*x**12 - 2522
52*a**3*b**5*x**15 - 140140*a**2*b**6*x**18 - 45045*a*b**7*x**21 - 6435*b**8*x**24)/(135135*x**45)

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