3.312 \(\int \frac{(a+b x^3)^8}{x^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{28}{17} a^2 b^6 x^{17}+4 a^3 b^5 x^{14}+\frac{70}{11} a^4 b^4 x^{11}+7 a^5 b^3 x^8+\frac{28}{5} a^6 b^2 x^5+4 a^7 b x^2-\frac{a^8}{x}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{23}}{23} \]

[Out]

-(a^8/x) + 4*a^7*b*x^2 + (28*a^6*b^2*x^5)/5 + 7*a^5*b^3*x^8 + (70*a^4*b^4*x^11)/11 + 4*a^3*b^5*x^14 + (28*a^2*
b^6*x^17)/17 + (2*a*b^7*x^20)/5 + (b^8*x^23)/23

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Rubi [A]  time = 0.0395976, antiderivative size = 100, 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} \[ \frac{28}{17} a^2 b^6 x^{17}+4 a^3 b^5 x^{14}+\frac{70}{11} a^4 b^4 x^{11}+7 a^5 b^3 x^8+\frac{28}{5} a^6 b^2 x^5+4 a^7 b x^2-\frac{a^8}{x}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^2,x]

[Out]

-(a^8/x) + 4*a^7*b*x^2 + (28*a^6*b^2*x^5)/5 + 7*a^5*b^3*x^8 + (70*a^4*b^4*x^11)/11 + 4*a^3*b^5*x^14 + (28*a^2*
b^6*x^17)/17 + (2*a*b^7*x^20)/5 + (b^8*x^23)/23

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^2} \, dx &=\int \left (\frac{a^8}{x^2}+8 a^7 b x+28 a^6 b^2 x^4+56 a^5 b^3 x^7+70 a^4 b^4 x^{10}+56 a^3 b^5 x^{13}+28 a^2 b^6 x^{16}+8 a b^7 x^{19}+b^8 x^{22}\right ) \, dx\\ &=-\frac{a^8}{x}+4 a^7 b x^2+\frac{28}{5} a^6 b^2 x^5+7 a^5 b^3 x^8+\frac{70}{11} a^4 b^4 x^{11}+4 a^3 b^5 x^{14}+\frac{28}{17} a^2 b^6 x^{17}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{23}}{23}\\ \end{align*}

Mathematica [A]  time = 0.0097131, size = 100, normalized size = 1. \[ \frac{28}{17} a^2 b^6 x^{17}+4 a^3 b^5 x^{14}+\frac{70}{11} a^4 b^4 x^{11}+7 a^5 b^3 x^8+\frac{28}{5} a^6 b^2 x^5+4 a^7 b x^2-\frac{a^8}{x}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^2,x]

[Out]

-(a^8/x) + 4*a^7*b*x^2 + (28*a^6*b^2*x^5)/5 + 7*a^5*b^3*x^8 + (70*a^4*b^4*x^11)/11 + 4*a^3*b^5*x^14 + (28*a^2*
b^6*x^17)/17 + (2*a*b^7*x^20)/5 + (b^8*x^23)/23

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Maple [A]  time = 0.003, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{x}}+4\,{a}^{7}b{x}^{2}+{\frac{28\,{a}^{6}{b}^{2}{x}^{5}}{5}}+7\,{a}^{5}{b}^{3}{x}^{8}+{\frac{70\,{a}^{4}{b}^{4}{x}^{11}}{11}}+4\,{a}^{3}{b}^{5}{x}^{14}+{\frac{28\,{a}^{2}{b}^{6}{x}^{17}}{17}}+{\frac{2\,a{b}^{7}{x}^{20}}{5}}+{\frac{{b}^{8}{x}^{23}}{23}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^2,x)

[Out]

-a^8/x+4*a^7*b*x^2+28/5*a^6*b^2*x^5+7*a^5*b^3*x^8+70/11*a^4*b^4*x^11+4*a^3*b^5*x^14+28/17*a^2*b^6*x^17+2/5*a*b
^7*x^20+1/23*b^8*x^23

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Maxima [A]  time = 0.949623, size = 122, normalized size = 1.22 \begin{align*} \frac{1}{23} \, b^{8} x^{23} + \frac{2}{5} \, a b^{7} x^{20} + \frac{28}{17} \, a^{2} b^{6} x^{17} + 4 \, a^{3} b^{5} x^{14} + \frac{70}{11} \, a^{4} b^{4} x^{11} + 7 \, a^{5} b^{3} x^{8} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 4 \, a^{7} b x^{2} - \frac{a^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="maxima")

[Out]

1/23*b^8*x^23 + 2/5*a*b^7*x^20 + 28/17*a^2*b^6*x^17 + 4*a^3*b^5*x^14 + 70/11*a^4*b^4*x^11 + 7*a^5*b^3*x^8 + 28
/5*a^6*b^2*x^5 + 4*a^7*b*x^2 - a^8/x

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Fricas [A]  time = 1.87744, size = 243, normalized size = 2.43 \begin{align*} \frac{935 \, b^{8} x^{24} + 8602 \, a b^{7} x^{21} + 35420 \, a^{2} b^{6} x^{18} + 86020 \, a^{3} b^{5} x^{15} + 136850 \, a^{4} b^{4} x^{12} + 150535 \, a^{5} b^{3} x^{9} + 120428 \, a^{6} b^{2} x^{6} + 86020 \, a^{7} b x^{3} - 21505 \, a^{8}}{21505 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="fricas")

[Out]

1/21505*(935*b^8*x^24 + 8602*a*b^7*x^21 + 35420*a^2*b^6*x^18 + 86020*a^3*b^5*x^15 + 136850*a^4*b^4*x^12 + 1505
35*a^5*b^3*x^9 + 120428*a^6*b^2*x^6 + 86020*a^7*b*x^3 - 21505*a^8)/x

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Sympy [A]  time = 0.431754, size = 99, normalized size = 0.99 \begin{align*} - \frac{a^{8}}{x} + 4 a^{7} b x^{2} + \frac{28 a^{6} b^{2} x^{5}}{5} + 7 a^{5} b^{3} x^{8} + \frac{70 a^{4} b^{4} x^{11}}{11} + 4 a^{3} b^{5} x^{14} + \frac{28 a^{2} b^{6} x^{17}}{17} + \frac{2 a b^{7} x^{20}}{5} + \frac{b^{8} x^{23}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**2,x)

[Out]

-a**8/x + 4*a**7*b*x**2 + 28*a**6*b**2*x**5/5 + 7*a**5*b**3*x**8 + 70*a**4*b**4*x**11/11 + 4*a**3*b**5*x**14 +
28*a**2*b**6*x**17/17 + 2*a*b**7*x**20/5 + b**8*x**23/23

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Giac [A]  time = 1.13904, size = 122, normalized size = 1.22 \begin{align*} \frac{1}{23} \, b^{8} x^{23} + \frac{2}{5} \, a b^{7} x^{20} + \frac{28}{17} \, a^{2} b^{6} x^{17} + 4 \, a^{3} b^{5} x^{14} + \frac{70}{11} \, a^{4} b^{4} x^{11} + 7 \, a^{5} b^{3} x^{8} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 4 \, a^{7} b x^{2} - \frac{a^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="giac")

[Out]

1/23*b^8*x^23 + 2/5*a*b^7*x^20 + 28/17*a^2*b^6*x^17 + 4*a^3*b^5*x^14 + 70/11*a^4*b^4*x^11 + 7*a^5*b^3*x^8 + 28
/5*a^6*b^2*x^5 + 4*a^7*b*x^2 - a^8/x