3.315 \(\int \frac{(a+b x^3)^8}{x^6} \, dx\)

Optimal. Leaf size=98 \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{28}{5} a^3 b^5 x^{10}+10 a^4 b^4 x^7+14 a^5 b^3 x^4+28 a^6 b^2 x-\frac{4 a^7 b}{x^2}-\frac{a^8}{5 x^5}+\frac{1}{2} a b^7 x^{16}+\frac{b^8 x^{19}}{19} \]

[Out]

-a^8/(5*x^5) - (4*a^7*b)/x^2 + 28*a^6*b^2*x + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 + (28*a^3*b^5*x^10)/5 + (28*a^2*
b^6*x^13)/13 + (a*b^7*x^16)/2 + (b^8*x^19)/19

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Rubi [A]  time = 0.0364757, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{28}{5} a^3 b^5 x^{10}+10 a^4 b^4 x^7+14 a^5 b^3 x^4+28 a^6 b^2 x-\frac{4 a^7 b}{x^2}-\frac{a^8}{5 x^5}+\frac{1}{2} a b^7 x^{16}+\frac{b^8 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^6,x]

[Out]

-a^8/(5*x^5) - (4*a^7*b)/x^2 + 28*a^6*b^2*x + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 + (28*a^3*b^5*x^10)/5 + (28*a^2*
b^6*x^13)/13 + (a*b^7*x^16)/2 + (b^8*x^19)/19

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^6} \, dx &=\int \left (28 a^6 b^2+\frac{a^8}{x^6}+\frac{8 a^7 b}{x^3}+56 a^5 b^3 x^3+70 a^4 b^4 x^6+56 a^3 b^5 x^9+28 a^2 b^6 x^{12}+8 a b^7 x^{15}+b^8 x^{18}\right ) \, dx\\ &=-\frac{a^8}{5 x^5}-\frac{4 a^7 b}{x^2}+28 a^6 b^2 x+14 a^5 b^3 x^4+10 a^4 b^4 x^7+\frac{28}{5} a^3 b^5 x^{10}+\frac{28}{13} a^2 b^6 x^{13}+\frac{1}{2} a b^7 x^{16}+\frac{b^8 x^{19}}{19}\\ \end{align*}

Mathematica [A]  time = 0.0096768, size = 98, normalized size = 1. \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{28}{5} a^3 b^5 x^{10}+10 a^4 b^4 x^7+14 a^5 b^3 x^4+28 a^6 b^2 x-\frac{4 a^7 b}{x^2}-\frac{a^8}{5 x^5}+\frac{1}{2} a b^7 x^{16}+\frac{b^8 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^6,x]

[Out]

-a^8/(5*x^5) - (4*a^7*b)/x^2 + 28*a^6*b^2*x + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 + (28*a^3*b^5*x^10)/5 + (28*a^2*
b^6*x^13)/13 + (a*b^7*x^16)/2 + (b^8*x^19)/19

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Maple [A]  time = 0.006, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{5\,{x}^{5}}}-4\,{\frac{{a}^{7}b}{{x}^{2}}}+28\,{a}^{6}{b}^{2}x+14\,{a}^{5}{b}^{3}{x}^{4}+10\,{a}^{4}{b}^{4}{x}^{7}+{\frac{28\,{a}^{3}{b}^{5}{x}^{10}}{5}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{13}}{13}}+{\frac{a{b}^{7}{x}^{16}}{2}}+{\frac{{b}^{8}{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^6,x)

[Out]

-1/5*a^8/x^5-4*a^7*b/x^2+28*a^6*b^2*x+14*a^5*b^3*x^4+10*a^4*b^4*x^7+28/5*a^3*b^5*x^10+28/13*a^2*b^6*x^13+1/2*a
*b^7*x^16+1/19*b^8*x^19

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Maxima [A]  time = 0.972571, size = 120, normalized size = 1.22 \begin{align*} \frac{1}{19} \, b^{8} x^{19} + \frac{1}{2} \, a b^{7} x^{16} + \frac{28}{13} \, a^{2} b^{6} x^{13} + \frac{28}{5} \, a^{3} b^{5} x^{10} + 10 \, a^{4} b^{4} x^{7} + 14 \, a^{5} b^{3} x^{4} + 28 \, a^{6} b^{2} x - \frac{20 \, a^{7} b x^{3} + a^{8}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="maxima")

[Out]

1/19*b^8*x^19 + 1/2*a*b^7*x^16 + 28/13*a^2*b^6*x^13 + 28/5*a^3*b^5*x^10 + 10*a^4*b^4*x^7 + 14*a^5*b^3*x^4 + 28
*a^6*b^2*x - 1/5*(20*a^7*b*x^3 + a^8)/x^5

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Fricas [A]  time = 1.68551, size = 235, normalized size = 2.4 \begin{align*} \frac{130 \, b^{8} x^{24} + 1235 \, a b^{7} x^{21} + 5320 \, a^{2} b^{6} x^{18} + 13832 \, a^{3} b^{5} x^{15} + 24700 \, a^{4} b^{4} x^{12} + 34580 \, a^{5} b^{3} x^{9} + 69160 \, a^{6} b^{2} x^{6} - 9880 \, a^{7} b x^{3} - 494 \, a^{8}}{2470 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="fricas")

[Out]

1/2470*(130*b^8*x^24 + 1235*a*b^7*x^21 + 5320*a^2*b^6*x^18 + 13832*a^3*b^5*x^15 + 24700*a^4*b^4*x^12 + 34580*a
^5*b^3*x^9 + 69160*a^6*b^2*x^6 - 9880*a^7*b*x^3 - 494*a^8)/x^5

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Sympy [A]  time = 0.418085, size = 97, normalized size = 0.99 \begin{align*} 28 a^{6} b^{2} x + 14 a^{5} b^{3} x^{4} + 10 a^{4} b^{4} x^{7} + \frac{28 a^{3} b^{5} x^{10}}{5} + \frac{28 a^{2} b^{6} x^{13}}{13} + \frac{a b^{7} x^{16}}{2} + \frac{b^{8} x^{19}}{19} - \frac{a^{8} + 20 a^{7} b x^{3}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**6,x)

[Out]

28*a**6*b**2*x + 14*a**5*b**3*x**4 + 10*a**4*b**4*x**7 + 28*a**3*b**5*x**10/5 + 28*a**2*b**6*x**13/13 + a*b**7
*x**16/2 + b**8*x**19/19 - (a**8 + 20*a**7*b*x**3)/(5*x**5)

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Giac [A]  time = 1.10661, size = 120, normalized size = 1.22 \begin{align*} \frac{1}{19} \, b^{8} x^{19} + \frac{1}{2} \, a b^{7} x^{16} + \frac{28}{13} \, a^{2} b^{6} x^{13} + \frac{28}{5} \, a^{3} b^{5} x^{10} + 10 \, a^{4} b^{4} x^{7} + 14 \, a^{5} b^{3} x^{4} + 28 \, a^{6} b^{2} x - \frac{20 \, a^{7} b x^{3} + a^{8}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="giac")

[Out]

1/19*b^8*x^19 + 1/2*a*b^7*x^16 + 28/13*a^2*b^6*x^13 + 28/5*a^3*b^5*x^10 + 10*a^4*b^4*x^7 + 14*a^5*b^3*x^4 + 28
*a^6*b^2*x - 1/5*(20*a^7*b*x^3 + a^8)/x^5