3.305 \(\int \frac{(a+b x^3)^8}{x^{40}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{\left (a+b x^3\right )^9}{39 a x^{39}} \]

[Out]

-(a + b*x^3)^9/(39*a*x^39) + (b*(a + b*x^3)^9)/(117*a^2*x^36) - (b^2*(a + b*x^3)^9)/(429*a^3*x^33) + (b^3*(a +
b*x^3)^9)/(2145*a^4*x^30) - (b^4*(a + b*x^3)^9)/(19305*a^5*x^27)

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Rubi [A]  time = 0.0521462, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ -\frac{b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{\left (a+b x^3\right )^9}{39 a x^{39}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^40,x]

[Out]

-(a + b*x^3)^9/(39*a*x^39) + (b*(a + b*x^3)^9)/(117*a^2*x^36) - (b^2*(a + b*x^3)^9)/(429*a^3*x^33) + (b^3*(a +
b*x^3)^9)/(2145*a^4*x^30) - (b^4*(a + b*x^3)^9)/(19305*a^5*x^27)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^{40}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{14}} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^9}{39 a x^{39}}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^3\right )}{39 a}\\ &=-\frac{\left (a+b x^3\right )^9}{39 a x^{39}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^3\right )}{39 a^2}\\ &=-\frac{\left (a+b x^3\right )^9}{39 a x^{39}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{429 a^3}\\ &=-\frac{\left (a+b x^3\right )^9}{39 a x^{39}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}+\frac{b^4 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{2145 a^4}\\ &=-\frac{\left (a+b x^3\right )^9}{39 a x^{39}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac{b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}\\ \end{align*}

Mathematica [A]  time = 0.0068878, size = 108, normalized size = 1.02 \[ -\frac{28 a^6 b^2}{33 x^{33}}-\frac{28 a^5 b^3}{15 x^{30}}-\frac{70 a^4 b^4}{27 x^{27}}-\frac{7 a^3 b^5}{3 x^{24}}-\frac{4 a^2 b^6}{3 x^{21}}-\frac{2 a^7 b}{9 x^{36}}-\frac{a^8}{39 x^{39}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^40,x]

[Out]

-a^8/(39*x^39) - (2*a^7*b)/(9*x^36) - (28*a^6*b^2)/(33*x^33) - (28*a^5*b^3)/(15*x^30) - (70*a^4*b^4)/(27*x^27)
- (7*a^3*b^5)/(3*x^24) - (4*a^2*b^6)/(3*x^21) - (4*a*b^7)/(9*x^18) - b^8/(15*x^15)

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Maple [A]  time = 0.007, size = 91, normalized size = 0.9 \begin{align*} -{\frac{28\,{a}^{6}{b}^{2}}{33\,{x}^{33}}}-{\frac{7\,{a}^{3}{b}^{5}}{3\,{x}^{24}}}-{\frac{4\,a{b}^{7}}{9\,{x}^{18}}}-{\frac{{a}^{8}}{39\,{x}^{39}}}-{\frac{70\,{a}^{4}{b}^{4}}{27\,{x}^{27}}}-{\frac{28\,{a}^{5}{b}^{3}}{15\,{x}^{30}}}-{\frac{{b}^{8}}{15\,{x}^{15}}}-{\frac{2\,{a}^{7}b}{9\,{x}^{36}}}-{\frac{4\,{a}^{2}{b}^{6}}{3\,{x}^{21}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^40,x)

[Out]

-28/33*a^6*b^2/x^33-7/3*a^3*b^5/x^24-4/9*a*b^7/x^18-1/39*a^8/x^39-70/27*a^4*b^4/x^27-28/15*a^5*b^3/x^30-1/15*b
^8/x^15-2/9*a^7*b/x^36-4/3*a^2*b^6/x^21

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Maxima [A]  time = 0.950414, size = 124, normalized size = 1.17 \begin{align*} -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^40,x, algorithm="maxima")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x^15 + 50050*a^4*b^4*x^12 + 360
36*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^3 + 495*a^8)/x^39

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Fricas [A]  time = 1.59567, size = 242, normalized size = 2.28 \begin{align*} -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^40,x, algorithm="fricas")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x^15 + 50050*a^4*b^4*x^12 + 360
36*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^3 + 495*a^8)/x^39

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Sympy [A]  time = 2.25843, size = 99, normalized size = 0.93 \begin{align*} - \frac{495 a^{8} + 4290 a^{7} b x^{3} + 16380 a^{6} b^{2} x^{6} + 36036 a^{5} b^{3} x^{9} + 50050 a^{4} b^{4} x^{12} + 45045 a^{3} b^{5} x^{15} + 25740 a^{2} b^{6} x^{18} + 8580 a b^{7} x^{21} + 1287 b^{8} x^{24}}{19305 x^{39}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**40,x)

[Out]

-(495*a**8 + 4290*a**7*b*x**3 + 16380*a**6*b**2*x**6 + 36036*a**5*b**3*x**9 + 50050*a**4*b**4*x**12 + 45045*a*
*3*b**5*x**15 + 25740*a**2*b**6*x**18 + 8580*a*b**7*x**21 + 1287*b**8*x**24)/(19305*x**39)

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Giac [A]  time = 1.12255, size = 124, normalized size = 1.17 \begin{align*} -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^40,x, algorithm="giac")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x^15 + 50050*a^4*b^4*x^12 + 360
36*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^3 + 495*a^8)/x^39