3.299 \(\int \frac{(a+b x^3)^8}{x^{22}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}-\frac{4 a^7 b}{9 x^{18}}-\frac{a^8}{21 x^{21}}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x^12) - (70*a^4*b^4)/(9*x^9) -
(28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8*x^3)/3 + 8*a*b^7*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0538065, antiderivative size = 105, 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} \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}-\frac{4 a^7 b}{9 x^{18}}-\frac{a^8}{21 x^{21}}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^22,x]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x^12) - (70*a^4*b^4)/(9*x^9) -
(28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8*x^3)/3 + 8*a*b^7*Log[x]

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^{22}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^8} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (b^8+\frac{a^8}{x^8}+\frac{8 a^7 b}{x^7}+\frac{28 a^6 b^2}{x^6}+\frac{56 a^5 b^3}{x^5}+\frac{70 a^4 b^4}{x^4}+\frac{56 a^3 b^5}{x^3}+\frac{28 a^2 b^6}{x^2}+\frac{8 a b^7}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{21 x^{21}}-\frac{4 a^7 b}{9 x^{18}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}+\frac{b^8 x^3}{3}+8 a b^7 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0103324, size = 105, normalized size = 1. \[ -\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}-\frac{4 a^7 b}{9 x^{18}}-\frac{a^8}{21 x^{21}}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^22,x]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x^12) - (70*a^4*b^4)/(9*x^9) -
(28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8*x^3)/3 + 8*a*b^7*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{21\,{x}^{21}}}-{\frac{4\,{a}^{7}b}{9\,{x}^{18}}}-{\frac{28\,{a}^{6}{b}^{2}}{15\,{x}^{15}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,{x}^{12}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,{x}^{9}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,{x}^{6}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}+{\frac{{b}^{8}{x}^{3}}{3}}+8\,a{b}^{7}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^22,x)

[Out]

-1/21*a^8/x^21-4/9*a^7*b/x^18-28/15*a^6*b^2/x^15-14/3*a^5*b^3/x^12-70/9*a^4*b^4/x^9-28/3*a^3*b^5/x^6-28/3*a^2*
b^6/x^3+1/3*b^8*x^3+8*a*b^7*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.967701, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{3} \, b^{8} x^{3} + \frac{8}{3} \, a b^{7} \log \left (x^{3}\right ) - \frac{2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^22,x, algorithm="maxima")

[Out]

1/3*b^8*x^3 + 8/3*a*b^7*log(x^3) - 1/315*(2940*a^2*b^6*x^18 + 2940*a^3*b^5*x^15 + 2450*a^4*b^4*x^12 + 1470*a^5
*b^3*x^9 + 588*a^6*b^2*x^6 + 140*a^7*b*x^3 + 15*a^8)/x^21

________________________________________________________________________________________

Fricas [A]  time = 1.64548, size = 235, normalized size = 2.24 \begin{align*} \frac{105 \, b^{8} x^{24} + 2520 \, a b^{7} x^{21} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{18} - 2940 \, a^{3} b^{5} x^{15} - 2450 \, a^{4} b^{4} x^{12} - 1470 \, a^{5} b^{3} x^{9} - 588 \, a^{6} b^{2} x^{6} - 140 \, a^{7} b x^{3} - 15 \, a^{8}}{315 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^22,x, algorithm="fricas")

[Out]

1/315*(105*b^8*x^24 + 2520*a*b^7*x^21*log(x) - 2940*a^2*b^6*x^18 - 2940*a^3*b^5*x^15 - 2450*a^4*b^4*x^12 - 147
0*a^5*b^3*x^9 - 588*a^6*b^2*x^6 - 140*a^7*b*x^3 - 15*a^8)/x^21

________________________________________________________________________________________

Sympy [A]  time = 1.42041, size = 97, normalized size = 0.92 \begin{align*} 8 a b^{7} \log{\left (x \right )} + \frac{b^{8} x^{3}}{3} - \frac{15 a^{8} + 140 a^{7} b x^{3} + 588 a^{6} b^{2} x^{6} + 1470 a^{5} b^{3} x^{9} + 2450 a^{4} b^{4} x^{12} + 2940 a^{3} b^{5} x^{15} + 2940 a^{2} b^{6} x^{18}}{315 x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**22,x)

[Out]

8*a*b**7*log(x) + b**8*x**3/3 - (15*a**8 + 140*a**7*b*x**3 + 588*a**6*b**2*x**6 + 1470*a**5*b**3*x**9 + 2450*a
**4*b**4*x**12 + 2940*a**3*b**5*x**15 + 2940*a**2*b**6*x**18)/(315*x**21)

________________________________________________________________________________________

Giac [A]  time = 1.11203, size = 138, normalized size = 1.31 \begin{align*} \frac{1}{3} \, b^{8} x^{3} + 8 \, a b^{7} \log \left ({\left | x \right |}\right ) - \frac{2178 \, a b^{7} x^{21} + 2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^22,x, algorithm="giac")

[Out]

1/3*b^8*x^3 + 8*a*b^7*log(abs(x)) - 1/315*(2178*a*b^7*x^21 + 2940*a^2*b^6*x^18 + 2940*a^3*b^5*x^15 + 2450*a^4*
b^4*x^12 + 1470*a^5*b^3*x^9 + 588*a^6*b^2*x^6 + 140*a^7*b*x^3 + 15*a^8)/x^21