3.100 \(\int \frac{(a+b x^2)^8}{x^{17}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a^7 b}{7 x^{14}}-\frac{a^8}{16 x^{16}}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^10) - (35*a^4*b^4)/(4*x^8) - (2
8*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + b^8*Log[x]

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Rubi [A]  time = 0.0540685, antiderivative size = 100, 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} \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a^7 b}{7 x^{14}}-\frac{a^8}{16 x^{16}}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^17,x]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^10) - (35*a^4*b^4)/(4*x^8) - (2
8*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + 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^2\right )^8}{x^{17}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^9} \, dx,x,x^2\right )\\ &=\frac{1}{2} \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^2\right )\\ &=-\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0045836, size = 100, normalized size = 1. \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a^7 b}{7 x^{14}}-\frac{a^8}{16 x^{16}}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^17,x]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^10) - (35*a^4*b^4)/(4*x^8) - (2
8*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + b^8*Log[x]

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Maple [A]  time = 0.008, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{16\,{x}^{16}}}-{\frac{4\,{a}^{7}b}{7\,{x}^{14}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,{x}^{12}}}-{\frac{28\,{a}^{5}{b}^{3}}{5\,{x}^{10}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,{x}^{8}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,{x}^{6}}}-7\,{\frac{{a}^{2}{b}^{6}}{{x}^{4}}}-4\,{\frac{a{b}^{7}}{{x}^{2}}}+{b}^{8}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^17,x)

[Out]

-1/16*a^8/x^16-4/7*a^7*b/x^14-7/3*a^6*b^2/x^12-28/5*a^5*b^3/x^10-35/4*a^4*b^4/x^8-28/3*a^3*b^5/x^6-7*a^2*b^6/x
^4-4*a*b^7/x^2+b^8*ln(x)

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Maxima [A]  time = 1.48186, size = 127, normalized size = 1.27 \begin{align*} \frac{1}{2} \, b^{8} \log \left (x^{2}\right ) - \frac{6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^17,x, algorithm="maxima")

[Out]

1/2*b^8*log(x^2) - 1/1680*(6720*a*b^7*x^14 + 11760*a^2*b^6*x^12 + 15680*a^3*b^5*x^10 + 14700*a^4*b^4*x^8 + 940
8*a^5*b^3*x^6 + 3920*a^6*b^2*x^4 + 960*a^7*b*x^2 + 105*a^8)/x^16

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Fricas [A]  time = 1.35895, size = 243, normalized size = 2.43 \begin{align*} \frac{1680 \, b^{8} x^{16} \log \left (x\right ) - 6720 \, a b^{7} x^{14} - 11760 \, a^{2} b^{6} x^{12} - 15680 \, a^{3} b^{5} x^{10} - 14700 \, a^{4} b^{4} x^{8} - 9408 \, a^{5} b^{3} x^{6} - 3920 \, a^{6} b^{2} x^{4} - 960 \, a^{7} b x^{2} - 105 \, a^{8}}{1680 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^17,x, algorithm="fricas")

[Out]

1/1680*(1680*b^8*x^16*log(x) - 6720*a*b^7*x^14 - 11760*a^2*b^6*x^12 - 15680*a^3*b^5*x^10 - 14700*a^4*b^4*x^8 -
9408*a^5*b^3*x^6 - 3920*a^6*b^2*x^4 - 960*a^7*b*x^2 - 105*a^8)/x^16

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Sympy [A]  time = 1.02748, size = 95, normalized size = 0.95 \begin{align*} b^{8} \log{\left (x \right )} - \frac{105 a^{8} + 960 a^{7} b x^{2} + 3920 a^{6} b^{2} x^{4} + 9408 a^{5} b^{3} x^{6} + 14700 a^{4} b^{4} x^{8} + 15680 a^{3} b^{5} x^{10} + 11760 a^{2} b^{6} x^{12} + 6720 a b^{7} x^{14}}{1680 x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**17,x)

[Out]

b**8*log(x) - (105*a**8 + 960*a**7*b*x**2 + 3920*a**6*b**2*x**4 + 9408*a**5*b**3*x**6 + 14700*a**4*b**4*x**8 +
15680*a**3*b**5*x**10 + 11760*a**2*b**6*x**12 + 6720*a*b**7*x**14)/(1680*x**16)

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Giac [A]  time = 2.51173, size = 138, normalized size = 1.38 \begin{align*} \frac{1}{2} \, b^{8} \log \left (x^{2}\right ) - \frac{2283 \, b^{8} x^{16} + 6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^17,x, algorithm="giac")

[Out]

1/2*b^8*log(x^2) - 1/1680*(2283*b^8*x^16 + 6720*a*b^7*x^14 + 11760*a^2*b^6*x^12 + 15680*a^3*b^5*x^10 + 14700*a
^4*b^4*x^8 + 9408*a^5*b^3*x^6 + 3920*a^6*b^2*x^4 + 960*a^7*b*x^2 + 105*a^8)/x^16