3.294 \(\int \frac{(a+b x^3)^8}{x^7} \, dx\)

Optimal. Leaf size=105 \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x
^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^6*b^2*Log[x]

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Rubi [A]  time = 0.0584267, 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} \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x
^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^6*b^2*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^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (56 a^5 b^3+\frac{a^8}{x^3}+\frac{8 a^7 b}{x^2}+\frac{28 a^6 b^2}{x}+70 a^4 b^4 x+56 a^3 b^5 x^2+28 a^2 b^6 x^3+8 a b^7 x^4+b^8 x^5\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{6 x^6}-\frac{8 a^7 b}{3 x^3}+\frac{56}{3} a^5 b^3 x^3+\frac{35}{3} a^4 b^4 x^6+\frac{56}{9} a^3 b^5 x^9+\frac{7}{3} a^2 b^6 x^{12}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18}+28 a^6 b^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0049907, size = 105, normalized size = 1. \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x
^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^6*b^2*Log[x]

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Maple [A]  time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{6\,{x}^{6}}}-{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{12}}{3}}+{\frac{8\,a{b}^{7}{x}^{15}}{15}}+{\frac{{b}^{8}{x}^{18}}{18}}+28\,{a}^{6}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^7,x)

[Out]

-1/6*a^8/x^6-8/3*a^7*b/x^3+56/3*a^5*b^3*x^3+35/3*a^4*b^4*x^6+56/9*a^3*b^5*x^9+7/3*a^2*b^6*x^12+8/15*a*b^7*x^15
+1/18*b^8*x^18+28*a^6*b^2*ln(x)

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Maxima [A]  time = 0.983206, size = 124, normalized size = 1.18 \begin{align*} \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + \frac{28}{3} \, a^{6} b^{2} \log \left (x^{3}\right ) - \frac{16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="maxima")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4*b^4*x^6 + 56/3*a^5*b^3*x^3 +
28/3*a^6*b^2*log(x^3) - 1/6*(16*a^7*b*x^3 + a^8)/x^6

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Fricas [A]  time = 1.58322, size = 225, normalized size = 2.14 \begin{align*} \frac{5 \, b^{8} x^{24} + 48 \, a b^{7} x^{21} + 210 \, a^{2} b^{6} x^{18} + 560 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 1680 \, a^{5} b^{3} x^{9} + 2520 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 240 \, a^{7} b x^{3} - 15 \, a^{8}}{90 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="fricas")

[Out]

1/90*(5*b^8*x^24 + 48*a*b^7*x^21 + 210*a^2*b^6*x^18 + 560*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 1680*a^5*b^3*x^9
+ 2520*a^6*b^2*x^6*log(x) - 240*a^7*b*x^3 - 15*a^8)/x^6

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Sympy [A]  time = 0.591675, size = 104, normalized size = 0.99 \begin{align*} 28 a^{6} b^{2} \log{\left (x \right )} + \frac{56 a^{5} b^{3} x^{3}}{3} + \frac{35 a^{4} b^{4} x^{6}}{3} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{8 a b^{7} x^{15}}{15} + \frac{b^{8} x^{18}}{18} - \frac{a^{8} + 16 a^{7} b x^{3}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**7,x)

[Out]

28*a**6*b**2*log(x) + 56*a**5*b**3*x**3/3 + 35*a**4*b**4*x**6/3 + 56*a**3*b**5*x**9/9 + 7*a**2*b**6*x**12/3 +
8*a*b**7*x**15/15 + b**8*x**18/18 - (a**8 + 16*a**7*b*x**3)/(6*x**6)

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Giac [A]  time = 1.62545, size = 138, normalized size = 1.31 \begin{align*} \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} \log \left ({\left | x \right |}\right ) - \frac{84 \, a^{6} b^{2} x^{6} + 16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="giac")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4*b^4*x^6 + 56/3*a^5*b^3*x^3 +
28*a^6*b^2*log(abs(x)) - 1/6*(84*a^6*b^2*x^6 + 16*a^7*b*x^3 + a^8)/x^6