3.306 \(\int \frac{(a+b x^3)^8}{x^{43}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{7 a^6 b^2}{9 x^{36}}-\frac{56 a^5 b^3}{33 x^{33}}-\frac{7 a^4 b^4}{3 x^{30}}-\frac{56 a^3 b^5}{27 x^{27}}-\frac{7 a^2 b^6}{6 x^{24}}-\frac{8 a^7 b}{39 x^{39}}-\frac{a^8}{42 x^{42}}-\frac{8 a b^7}{21 x^{21}}-\frac{b^8}{18 x^{18}} \]

[Out]

-a^8/(42*x^42) - (8*a^7*b)/(39*x^39) - (7*a^6*b^2)/(9*x^36) - (56*a^5*b^3)/(33*x^33) - (7*a^4*b^4)/(3*x^30) -
(56*a^3*b^5)/(27*x^27) - (7*a^2*b^6)/(6*x^24) - (8*a*b^7)/(21*x^21) - b^8/(18*x^18)

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Rubi [A]  time = 0.05092, 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} \[ -\frac{7 a^6 b^2}{9 x^{36}}-\frac{56 a^5 b^3}{33 x^{33}}-\frac{7 a^4 b^4}{3 x^{30}}-\frac{56 a^3 b^5}{27 x^{27}}-\frac{7 a^2 b^6}{6 x^{24}}-\frac{8 a^7 b}{39 x^{39}}-\frac{a^8}{42 x^{42}}-\frac{8 a b^7}{21 x^{21}}-\frac{b^8}{18 x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^43,x]

[Out]

-a^8/(42*x^42) - (8*a^7*b)/(39*x^39) - (7*a^6*b^2)/(9*x^36) - (56*a^5*b^3)/(33*x^33) - (7*a^4*b^4)/(3*x^30) -
(56*a^3*b^5)/(27*x^27) - (7*a^2*b^6)/(6*x^24) - (8*a*b^7)/(21*x^21) - b^8/(18*x^18)

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^{43}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{15}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^{15}}+\frac{8 a^7 b}{x^{14}}+\frac{28 a^6 b^2}{x^{13}}+\frac{56 a^5 b^3}{x^{12}}+\frac{70 a^4 b^4}{x^{11}}+\frac{56 a^3 b^5}{x^{10}}+\frac{28 a^2 b^6}{x^9}+\frac{8 a b^7}{x^8}+\frac{b^8}{x^7}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{42 x^{42}}-\frac{8 a^7 b}{39 x^{39}}-\frac{7 a^6 b^2}{9 x^{36}}-\frac{56 a^5 b^3}{33 x^{33}}-\frac{7 a^4 b^4}{3 x^{30}}-\frac{56 a^3 b^5}{27 x^{27}}-\frac{7 a^2 b^6}{6 x^{24}}-\frac{8 a b^7}{21 x^{21}}-\frac{b^8}{18 x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0046637, size = 108, normalized size = 1. \[ -\frac{7 a^6 b^2}{9 x^{36}}-\frac{56 a^5 b^3}{33 x^{33}}-\frac{7 a^4 b^4}{3 x^{30}}-\frac{56 a^3 b^5}{27 x^{27}}-\frac{7 a^2 b^6}{6 x^{24}}-\frac{8 a^7 b}{39 x^{39}}-\frac{a^8}{42 x^{42}}-\frac{8 a b^7}{21 x^{21}}-\frac{b^8}{18 x^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^43,x]

[Out]

-a^8/(42*x^42) - (8*a^7*b)/(39*x^39) - (7*a^6*b^2)/(9*x^36) - (56*a^5*b^3)/(33*x^33) - (7*a^4*b^4)/(3*x^30) -
(56*a^3*b^5)/(27*x^27) - (7*a^2*b^6)/(6*x^24) - (8*a*b^7)/(21*x^21) - b^8/(18*x^18)

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Maple [A]  time = 0.007, size = 91, normalized size = 0.8 \begin{align*} -{\frac{{a}^{8}}{42\,{x}^{42}}}-{\frac{8\,{a}^{7}b}{39\,{x}^{39}}}-{\frac{7\,{a}^{6}{b}^{2}}{9\,{x}^{36}}}-{\frac{56\,{a}^{5}{b}^{3}}{33\,{x}^{33}}}-{\frac{7\,{a}^{4}{b}^{4}}{3\,{x}^{30}}}-{\frac{56\,{a}^{3}{b}^{5}}{27\,{x}^{27}}}-{\frac{7\,{a}^{2}{b}^{6}}{6\,{x}^{24}}}-{\frac{8\,a{b}^{7}}{21\,{x}^{21}}}-{\frac{{b}^{8}}{18\,{x}^{18}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^43,x)

[Out]

-1/42*a^8/x^42-8/39*a^7*b/x^39-7/9*a^6*b^2/x^36-56/33*a^5*b^3/x^33-7/3*a^4*b^4/x^30-56/27*a^3*b^5/x^27-7/6*a^2
*b^6/x^24-8/21*a*b^7/x^21-1/18*b^8/x^18

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Maxima [A]  time = 0.967416, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="maxima")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42

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Fricas [A]  time = 1.57446, size = 248, normalized size = 2.3 \begin{align*} -\frac{3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="fricas")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42

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Sympy [A]  time = 2.3996, size = 99, normalized size = 0.92 \begin{align*} - \frac{1287 a^{8} + 11088 a^{7} b x^{3} + 42042 a^{6} b^{2} x^{6} + 91728 a^{5} b^{3} x^{9} + 126126 a^{4} b^{4} x^{12} + 112112 a^{3} b^{5} x^{15} + 63063 a^{2} b^{6} x^{18} + 20592 a b^{7} x^{21} + 3003 b^{8} x^{24}}{54054 x^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**43,x)

[Out]

-(1287*a**8 + 11088*a**7*b*x**3 + 42042*a**6*b**2*x**6 + 91728*a**5*b**3*x**9 + 126126*a**4*b**4*x**12 + 11211
2*a**3*b**5*x**15 + 63063*a**2*b**6*x**18 + 20592*a*b**7*x**21 + 3003*b**8*x**24)/(54054*x**42)

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Giac [A]  time = 1.10191, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="giac")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42