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

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

[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)

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Rubi [A]  time = 0.0492615, antiderivative size = 108, 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, 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 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]]

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])

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*}

Mathematica [A]  time = 0.0097878, size = 108, normalized size = 1. \[ -\frac{28 a^6 b^2}{39 x^{39}}-\frac{14 a^5 b^3}{9 x^{36}}-\frac{70 a^4 b^4}{33 x^{33}}-\frac{28 a^3 b^5}{15 x^{30}}-\frac{28 a^2 b^6}{27 x^{27}}-\frac{4 a^7 b}{21 x^{42}}-\frac{a^8}{45 x^{45}}-\frac{a b^7}{3 x^{24}}-\frac{b^8}{21 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(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)

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Maple [A]  time = 0.007, size = 91, normalized size = 0.8 \begin{align*} -{\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*}

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 = 0.979875, size = 124, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.66625, size = 252, normalized size = 2.33 \begin{align*} -\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}} \end{align*}

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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Sympy [A]  time = 2.52528, size = 99, normalized size = 0.92 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.1074, size = 124, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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