3.314 \(\int \frac{(a+b x^3)^8}{x^5} \, dx\)

Optimal. Leaf size=102 \[ 2 a^2 b^6 x^{14}+\frac{56}{11} a^3 b^5 x^{11}+\frac{35}{4} a^4 b^4 x^8+\frac{56}{5} a^5 b^3 x^5+14 a^6 b^2 x^2-\frac{8 a^7 b}{x}-\frac{a^8}{4 x^4}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{20}}{20} \]

[Out]

-a^8/(4*x^4) - (8*a^7*b)/x + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 +
2*a^2*b^6*x^14 + (8*a*b^7*x^17)/17 + (b^8*x^20)/20

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Rubi [A]  time = 0.0375903, antiderivative size = 102, 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} \[ 2 a^2 b^6 x^{14}+\frac{56}{11} a^3 b^5 x^{11}+\frac{35}{4} a^4 b^4 x^8+\frac{56}{5} a^5 b^3 x^5+14 a^6 b^2 x^2-\frac{8 a^7 b}{x}-\frac{a^8}{4 x^4}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^5,x]

[Out]

-a^8/(4*x^4) - (8*a^7*b)/x + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 +
2*a^2*b^6*x^14 + (8*a*b^7*x^17)/17 + (b^8*x^20)/20

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^5} \, dx &=\int \left (\frac{a^8}{x^5}+\frac{8 a^7 b}{x^2}+28 a^6 b^2 x+56 a^5 b^3 x^4+70 a^4 b^4 x^7+56 a^3 b^5 x^{10}+28 a^2 b^6 x^{13}+8 a b^7 x^{16}+b^8 x^{19}\right ) \, dx\\ &=-\frac{a^8}{4 x^4}-\frac{8 a^7 b}{x}+14 a^6 b^2 x^2+\frac{56}{5} a^5 b^3 x^5+\frac{35}{4} a^4 b^4 x^8+\frac{56}{11} a^3 b^5 x^{11}+2 a^2 b^6 x^{14}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{20}}{20}\\ \end{align*}

Mathematica [A]  time = 0.0046417, size = 102, normalized size = 1. \[ 2 a^2 b^6 x^{14}+\frac{56}{11} a^3 b^5 x^{11}+\frac{35}{4} a^4 b^4 x^8+\frac{56}{5} a^5 b^3 x^5+14 a^6 b^2 x^2-\frac{8 a^7 b}{x}-\frac{a^8}{4 x^4}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^5,x]

[Out]

-a^8/(4*x^4) - (8*a^7*b)/x + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 +
2*a^2*b^6*x^14 + (8*a*b^7*x^17)/17 + (b^8*x^20)/20

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Maple [A]  time = 0.005, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{4\,{x}^{4}}}-8\,{\frac{{a}^{7}b}{x}}+14\,{a}^{6}{b}^{2}{x}^{2}+{\frac{56\,{a}^{5}{b}^{3}{x}^{5}}{5}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{8}}{4}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{11}}{11}}+2\,{a}^{2}{b}^{6}{x}^{14}+{\frac{8\,a{b}^{7}{x}^{17}}{17}}+{\frac{{b}^{8}{x}^{20}}{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^5,x)

[Out]

-1/4*a^8/x^4-8*a^7*b/x+14*a^6*b^2*x^2+56/5*a^5*b^3*x^5+35/4*a^4*b^4*x^8+56/11*a^3*b^5*x^11+2*a^2*b^6*x^14+8/17
*a*b^7*x^17+1/20*b^8*x^20

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Maxima [A]  time = 0.953152, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{20} \, b^{8} x^{20} + \frac{8}{17} \, a b^{7} x^{17} + 2 \, a^{2} b^{6} x^{14} + \frac{56}{11} \, a^{3} b^{5} x^{11} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{56}{5} \, a^{5} b^{3} x^{5} + 14 \, a^{6} b^{2} x^{2} - \frac{32 \, a^{7} b x^{3} + a^{8}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="maxima")

[Out]

1/20*b^8*x^20 + 8/17*a*b^7*x^17 + 2*a^2*b^6*x^14 + 56/11*a^3*b^5*x^11 + 35/4*a^4*b^4*x^8 + 56/5*a^5*b^3*x^5 +
14*a^6*b^2*x^2 - 1/4*(32*a^7*b*x^3 + a^8)/x^4

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Fricas [A]  time = 1.69644, size = 236, normalized size = 2.31 \begin{align*} \frac{187 \, b^{8} x^{24} + 1760 \, a b^{7} x^{21} + 7480 \, a^{2} b^{6} x^{18} + 19040 \, a^{3} b^{5} x^{15} + 32725 \, a^{4} b^{4} x^{12} + 41888 \, a^{5} b^{3} x^{9} + 52360 \, a^{6} b^{2} x^{6} - 29920 \, a^{7} b x^{3} - 935 \, a^{8}}{3740 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="fricas")

[Out]

1/3740*(187*b^8*x^24 + 1760*a*b^7*x^21 + 7480*a^2*b^6*x^18 + 19040*a^3*b^5*x^15 + 32725*a^4*b^4*x^12 + 41888*a
^5*b^3*x^9 + 52360*a^6*b^2*x^6 - 29920*a^7*b*x^3 - 935*a^8)/x^4

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Sympy [A]  time = 0.511519, size = 102, normalized size = 1. \begin{align*} 14 a^{6} b^{2} x^{2} + \frac{56 a^{5} b^{3} x^{5}}{5} + \frac{35 a^{4} b^{4} x^{8}}{4} + \frac{56 a^{3} b^{5} x^{11}}{11} + 2 a^{2} b^{6} x^{14} + \frac{8 a b^{7} x^{17}}{17} + \frac{b^{8} x^{20}}{20} - \frac{a^{8} + 32 a^{7} b x^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**5,x)

[Out]

14*a**6*b**2*x**2 + 56*a**5*b**3*x**5/5 + 35*a**4*b**4*x**8/4 + 56*a**3*b**5*x**11/11 + 2*a**2*b**6*x**14 + 8*
a*b**7*x**17/17 + b**8*x**20/20 - (a**8 + 32*a**7*b*x**3)/(4*x**4)

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Giac [A]  time = 1.11894, size = 123, normalized size = 1.21 \begin{align*} \frac{1}{20} \, b^{8} x^{20} + \frac{8}{17} \, a b^{7} x^{17} + 2 \, a^{2} b^{6} x^{14} + \frac{56}{11} \, a^{3} b^{5} x^{11} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{56}{5} \, a^{5} b^{3} x^{5} + 14 \, a^{6} b^{2} x^{2} - \frac{32 \, a^{7} b x^{3} + a^{8}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="giac")

[Out]

1/20*b^8*x^20 + 8/17*a*b^7*x^17 + 2*a^2*b^6*x^14 + 56/11*a^3*b^5*x^11 + 35/4*a^4*b^4*x^8 + 56/5*a^5*b^3*x^5 +
14*a^6*b^2*x^2 - 1/4*(32*a^7*b*x^3 + a^8)/x^4