3.302 \(\int \frac{(a+b x^3)^8}{x^{31}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^3\right )^9}{270 a^2 x^{27}}-\frac{\left (a+b x^3\right )^9}{30 a x^{30}} \]

[Out]

-(a + b*x^3)^9/(30*a*x^30) + (b*(a + b*x^3)^9)/(270*a^2*x^27)

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Rubi [A]  time = 0.0176665, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b \left (a+b x^3\right )^9}{270 a^2 x^{27}}-\frac{\left (a+b x^3\right )^9}{30 a x^{30}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^31,x]

[Out]

-(a + b*x^3)^9/(30*a*x^30) + (b*(a + b*x^3)^9)/(270*a^2*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^{31}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^9}{30 a x^{30}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{30 a}\\ &=-\frac{\left (a+b x^3\right )^9}{30 a x^{30}}+\frac{b \left (a+b x^3\right )^9}{270 a^2 x^{27}}\\ \end{align*}

Mathematica [B]  time = 0.0049524, size = 108, normalized size = 2.7 \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{8 a^5 b^3}{3 x^{21}}-\frac{35 a^4 b^4}{9 x^{18}}-\frac{56 a^3 b^5}{15 x^{15}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a^7 b}{27 x^{27}}-\frac{a^8}{30 x^{30}}-\frac{8 a b^7}{9 x^9}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^31,x]

[Out]

-a^8/(30*x^30) - (8*a^7*b)/(27*x^27) - (7*a^6*b^2)/(6*x^24) - (8*a^5*b^3)/(3*x^21) - (35*a^4*b^4)/(9*x^18) - (
56*a^3*b^5)/(15*x^15) - (7*a^2*b^6)/(3*x^12) - (8*a*b^7)/(9*x^9) - b^8/(6*x^6)

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Maple [B]  time = 0.006, size = 91, normalized size = 2.3 \begin{align*} -{\frac{{a}^{8}}{30\,{x}^{30}}}-{\frac{56\,{a}^{3}{b}^{5}}{15\,{x}^{15}}}-{\frac{7\,{a}^{6}{b}^{2}}{6\,{x}^{24}}}-{\frac{7\,{a}^{2}{b}^{6}}{3\,{x}^{12}}}-{\frac{8\,{a}^{5}{b}^{3}}{3\,{x}^{21}}}-{\frac{35\,{a}^{4}{b}^{4}}{9\,{x}^{18}}}-{\frac{8\,{a}^{7}b}{27\,{x}^{27}}}-{\frac{{b}^{8}}{6\,{x}^{6}}}-{\frac{8\,a{b}^{7}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^31,x)

[Out]

-1/30*a^8/x^30-56/15*a^3*b^5/x^15-7/6*a^6*b^2/x^24-7/3*a^2*b^6/x^12-8/3*a^5*b^3/x^21-35/9*a^4*b^4/x^18-8/27*a^
7*b/x^27-1/6*b^8/x^6-8/9*a*b^7/x^9

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Maxima [B]  time = 0.98502, size = 124, normalized size = 3.1 \begin{align*} -\frac{45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="maxima")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30

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Fricas [B]  time = 1.58316, size = 219, normalized size = 5.48 \begin{align*} -\frac{45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="fricas")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30

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Sympy [B]  time = 1.93233, size = 99, normalized size = 2.48 \begin{align*} - \frac{9 a^{8} + 80 a^{7} b x^{3} + 315 a^{6} b^{2} x^{6} + 720 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12} + 1008 a^{3} b^{5} x^{15} + 630 a^{2} b^{6} x^{18} + 240 a b^{7} x^{21} + 45 b^{8} x^{24}}{270 x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**31,x)

[Out]

-(9*a**8 + 80*a**7*b*x**3 + 315*a**6*b**2*x**6 + 720*a**5*b**3*x**9 + 1050*a**4*b**4*x**12 + 1008*a**3*b**5*x*
*15 + 630*a**2*b**6*x**18 + 240*a*b**7*x**21 + 45*b**8*x**24)/(270*x**30)

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Giac [B]  time = 1.10768, size = 124, normalized size = 3.1 \begin{align*} -\frac{45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="giac")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30