3.300 \(\int \frac{(a+b x^3)^8}{x^{25}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*x^15) - (35*a^4*b^4)/(6*x^12)
- (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8*a*b^7)/(3*x^3) + b^8*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0507342, antiderivative size = 104, 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} \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^25,x]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*x^15) - (35*a^4*b^4)/(6*x^12)
- (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8*a*b^7)/(3*x^3) + b^8*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^{25}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^9} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^9}+\frac{8 a^7 b}{x^8}+\frac{28 a^6 b^2}{x^7}+\frac{56 a^5 b^3}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^3 b^5}{x^4}+\frac{28 a^2 b^6}{x^3}+\frac{8 a b^7}{x^2}+\frac{b^8}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{24 x^{24}}-\frac{8 a^7 b}{21 x^{21}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{3 x^3}+b^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0049155, size = 104, normalized size = 1. \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^25,x]

[Out]

-a^8/(24*x^24) - (8*a^7*b)/(21*x^21) - (14*a^6*b^2)/(9*x^18) - (56*a^5*b^3)/(15*x^15) - (35*a^4*b^4)/(6*x^12)
- (56*a^3*b^5)/(9*x^9) - (14*a^2*b^6)/(3*x^6) - (8*a*b^7)/(3*x^3) + b^8*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{24\,{x}^{24}}}-{\frac{8\,{a}^{7}b}{21\,{x}^{21}}}-{\frac{14\,{a}^{6}{b}^{2}}{9\,{x}^{18}}}-{\frac{56\,{a}^{5}{b}^{3}}{15\,{x}^{15}}}-{\frac{35\,{a}^{4}{b}^{4}}{6\,{x}^{12}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{8\,a{b}^{7}}{3\,{x}^{3}}}+{b}^{8}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^25,x)

[Out]

-1/24*a^8/x^24-8/21*a^7*b/x^21-14/9*a^6*b^2/x^18-56/15*a^5*b^3/x^15-35/6*a^4*b^4/x^12-56/9*a^3*b^5/x^9-14/3*a^
2*b^6/x^6-8/3*a*b^7/x^3+b^8*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.951666, size = 127, normalized size = 1.22 \begin{align*} \frac{1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac{6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^25,x, algorithm="maxima")

[Out]

1/3*b^8*log(x^3) - 1/2520*(6720*a*b^7*x^21 + 11760*a^2*b^6*x^18 + 15680*a^3*b^5*x^15 + 14700*a^4*b^4*x^12 + 94
08*a^5*b^3*x^9 + 3920*a^6*b^2*x^6 + 960*a^7*b*x^3 + 105*a^8)/x^24

________________________________________________________________________________________

Fricas [A]  time = 1.69963, size = 244, normalized size = 2.35 \begin{align*} \frac{2520 \, b^{8} x^{24} \log \left (x\right ) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^25,x, algorithm="fricas")

[Out]

1/2520*(2520*b^8*x^24*log(x) - 6720*a*b^7*x^21 - 11760*a^2*b^6*x^18 - 15680*a^3*b^5*x^15 - 14700*a^4*b^4*x^12
- 9408*a^5*b^3*x^9 - 3920*a^6*b^2*x^6 - 960*a^7*b*x^3 - 105*a^8)/x^24

________________________________________________________________________________________

Sympy [A]  time = 1.26163, size = 95, normalized size = 0.91 \begin{align*} b^{8} \log{\left (x \right )} - \frac{105 a^{8} + 960 a^{7} b x^{3} + 3920 a^{6} b^{2} x^{6} + 9408 a^{5} b^{3} x^{9} + 14700 a^{4} b^{4} x^{12} + 15680 a^{3} b^{5} x^{15} + 11760 a^{2} b^{6} x^{18} + 6720 a b^{7} x^{21}}{2520 x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**25,x)

[Out]

b**8*log(x) - (105*a**8 + 960*a**7*b*x**3 + 3920*a**6*b**2*x**6 + 9408*a**5*b**3*x**9 + 14700*a**4*b**4*x**12
+ 15680*a**3*b**5*x**15 + 11760*a**2*b**6*x**18 + 6720*a*b**7*x**21)/(2520*x**24)

________________________________________________________________________________________

Giac [A]  time = 1.14023, size = 135, normalized size = 1.3 \begin{align*} b^{8} \log \left ({\left | x \right |}\right ) - \frac{2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^25,x, algorithm="giac")

[Out]

b^8*log(abs(x)) - 1/2520*(2283*b^8*x^24 + 6720*a*b^7*x^21 + 11760*a^2*b^6*x^18 + 15680*a^3*b^5*x^15 + 14700*a^
4*b^4*x^12 + 9408*a^5*b^3*x^9 + 3920*a^6*b^2*x^6 + 960*a^7*b*x^3 + 105*a^8)/x^24