3.98 \(\int \frac{(a+b x^2)^8}{x^{13}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)-\frac{4 a^7 b}{5 x^{10}}-\frac{a^8}{12 x^{12}}+4 a b^7 x^2+\frac{b^8 x^4}{4} \]

[Out]

-a^8/(12*x^12) - (4*a^7*b)/(5*x^10) - (7*a^6*b^2)/(2*x^8) - (28*a^5*b^3)/(3*x^6) - (35*a^4*b^4)/(2*x^4) - (28*
a^3*b^5)/x^2 + 4*a*b^7*x^2 + (b^8*x^4)/4 + 28*a^2*b^6*Log[x]

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Rubi [A]  time = 0.0536426, antiderivative size = 101, 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} \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)-\frac{4 a^7 b}{5 x^{10}}-\frac{a^8}{12 x^{12}}+4 a b^7 x^2+\frac{b^8 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (4*a^7*b)/(5*x^10) - (7*a^6*b^2)/(2*x^8) - (28*a^5*b^3)/(3*x^6) - (35*a^4*b^4)/(2*x^4) - (28*
a^3*b^5)/x^2 + 4*a*b^7*x^2 + (b^8*x^4)/4 + 28*a^2*b^6*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^2\right )^8}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^7} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (8 a b^7+\frac{a^8}{x^7}+\frac{8 a^7 b}{x^6}+\frac{28 a^6 b^2}{x^5}+\frac{56 a^5 b^3}{x^4}+\frac{70 a^4 b^4}{x^3}+\frac{56 a^3 b^5}{x^2}+\frac{28 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{12 x^{12}}-\frac{4 a^7 b}{5 x^{10}}-\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+4 a b^7 x^2+\frac{b^8 x^4}{4}+28 a^2 b^6 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0045843, size = 101, normalized size = 1. \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)-\frac{4 a^7 b}{5 x^{10}}-\frac{a^8}{12 x^{12}}+4 a b^7 x^2+\frac{b^8 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (4*a^7*b)/(5*x^10) - (7*a^6*b^2)/(2*x^8) - (28*a^5*b^3)/(3*x^6) - (35*a^4*b^4)/(2*x^4) - (28*
a^3*b^5)/x^2 + 4*a*b^7*x^2 + (b^8*x^4)/4 + 28*a^2*b^6*Log[x]

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Maple [A]  time = 0.008, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{12\,{x}^{12}}}-{\frac{4\,{a}^{7}b}{5\,{x}^{10}}}-{\frac{7\,{a}^{6}{b}^{2}}{2\,{x}^{8}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,{x}^{4}}}-28\,{\frac{{a}^{3}{b}^{5}}{{x}^{2}}}+4\,a{b}^{7}{x}^{2}+{\frac{{b}^{8}{x}^{4}}{4}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^13,x)

[Out]

-1/12*a^8/x^12-4/5*a^7*b/x^10-7/2*a^6*b^2/x^8-28/3*a^5*b^3/x^6-35/2*a^4*b^4/x^4-28*a^3*b^5/x^2+4*a*b^7*x^2+1/4
*b^8*x^4+28*a^2*b^6*ln(x)

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Maxima [A]  time = 1.63539, size = 127, normalized size = 1.26 \begin{align*} \frac{1}{4} \, b^{8} x^{4} + 4 \, a b^{7} x^{2} + 14 \, a^{2} b^{6} \log \left (x^{2}\right ) - \frac{1680 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 560 \, a^{5} b^{3} x^{6} + 210 \, a^{6} b^{2} x^{4} + 48 \, a^{7} b x^{2} + 5 \, a^{8}}{60 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^13,x, algorithm="maxima")

[Out]

1/4*b^8*x^4 + 4*a*b^7*x^2 + 14*a^2*b^6*log(x^2) - 1/60*(1680*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 560*a^5*b^3*x^6
+ 210*a^6*b^2*x^4 + 48*a^7*b*x^2 + 5*a^8)/x^12

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Fricas [A]  time = 1.25541, size = 225, normalized size = 2.23 \begin{align*} \frac{15 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 1680 \, a^{2} b^{6} x^{12} \log \left (x\right ) - 1680 \, a^{3} b^{5} x^{10} - 1050 \, a^{4} b^{4} x^{8} - 560 \, a^{5} b^{3} x^{6} - 210 \, a^{6} b^{2} x^{4} - 48 \, a^{7} b x^{2} - 5 \, a^{8}}{60 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^13,x, algorithm="fricas")

[Out]

1/60*(15*b^8*x^16 + 240*a*b^7*x^14 + 1680*a^2*b^6*x^12*log(x) - 1680*a^3*b^5*x^10 - 1050*a^4*b^4*x^8 - 560*a^5
*b^3*x^6 - 210*a^6*b^2*x^4 - 48*a^7*b*x^2 - 5*a^8)/x^12

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Sympy [A]  time = 1.00977, size = 97, normalized size = 0.96 \begin{align*} 28 a^{2} b^{6} \log{\left (x \right )} + 4 a b^{7} x^{2} + \frac{b^{8} x^{4}}{4} - \frac{5 a^{8} + 48 a^{7} b x^{2} + 210 a^{6} b^{2} x^{4} + 560 a^{5} b^{3} x^{6} + 1050 a^{4} b^{4} x^{8} + 1680 a^{3} b^{5} x^{10}}{60 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**13,x)

[Out]

28*a**2*b**6*log(x) + 4*a*b**7*x**2 + b**8*x**4/4 - (5*a**8 + 48*a**7*b*x**2 + 210*a**6*b**2*x**4 + 560*a**5*b
**3*x**6 + 1050*a**4*b**4*x**8 + 1680*a**3*b**5*x**10)/(60*x**12)

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Giac [A]  time = 2.82687, size = 142, normalized size = 1.41 \begin{align*} \frac{1}{4} \, b^{8} x^{4} + 4 \, a b^{7} x^{2} + 14 \, a^{2} b^{6} \log \left (x^{2}\right ) - \frac{2058 \, a^{2} b^{6} x^{12} + 1680 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 560 \, a^{5} b^{3} x^{6} + 210 \, a^{6} b^{2} x^{4} + 48 \, a^{7} b x^{2} + 5 \, a^{8}}{60 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^13,x, algorithm="giac")

[Out]

1/4*b^8*x^4 + 4*a*b^7*x^2 + 14*a^2*b^6*log(x^2) - 1/60*(2058*a^2*b^6*x^12 + 1680*a^3*b^5*x^10 + 1050*a^4*b^4*x
^8 + 560*a^5*b^3*x^6 + 210*a^6*b^2*x^4 + 48*a^7*b*x^2 + 5*a^8)/x^12