3.105 \(\int \frac{(a+b x^2)^8}{x^{27}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{b^4 \left (a+b x^2\right )^9}{12870 a^5 x^{18}}+\frac{b^3 \left (a+b x^2\right )^9}{1430 a^4 x^{20}}-\frac{b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac{\left (a+b x^2\right )^9}{26 a x^{26}} \]

[Out]

-(a + b*x^2)^9/(26*a*x^26) + (b*(a + b*x^2)^9)/(78*a^2*x^24) - (b^2*(a + b*x^2)^9)/(286*a^3*x^22) + (b^3*(a +
b*x^2)^9)/(1430*a^4*x^20) - (b^4*(a + b*x^2)^9)/(12870*a^5*x^18)

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Rubi [A]  time = 0.054699, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ -\frac{b^4 \left (a+b x^2\right )^9}{12870 a^5 x^{18}}+\frac{b^3 \left (a+b x^2\right )^9}{1430 a^4 x^{20}}-\frac{b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac{\left (a+b x^2\right )^9}{26 a x^{26}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^27,x]

[Out]

-(a + b*x^2)^9/(26*a*x^26) + (b*(a + b*x^2)^9)/(78*a^2*x^24) - (b^2*(a + b*x^2)^9)/(286*a^3*x^22) + (b^3*(a +
b*x^2)^9)/(1430*a^4*x^20) - (b^4*(a + b*x^2)^9)/(12870*a^5*x^18)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^{27}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{14}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^9}{26 a x^{26}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^2\right )}{13 a}\\ &=-\frac{\left (a+b x^2\right )^9}{26 a x^{26}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^2\right )}{26 a^2}\\ &=-\frac{\left (a+b x^2\right )^9}{26 a x^{26}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac{b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^2\right )}{143 a^3}\\ &=-\frac{\left (a+b x^2\right )^9}{26 a x^{26}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac{b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}+\frac{b^3 \left (a+b x^2\right )^9}{1430 a^4 x^{20}}+\frac{b^4 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{1430 a^4}\\ &=-\frac{\left (a+b x^2\right )^9}{26 a x^{26}}+\frac{b \left (a+b x^2\right )^9}{78 a^2 x^{24}}-\frac{b^2 \left (a+b x^2\right )^9}{286 a^3 x^{22}}+\frac{b^3 \left (a+b x^2\right )^9}{1430 a^4 x^{20}}-\frac{b^4 \left (a+b x^2\right )^9}{12870 a^5 x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0043688, size = 106, normalized size = 1. \[ -\frac{14 a^6 b^2}{11 x^{22}}-\frac{14 a^5 b^3}{5 x^{20}}-\frac{35 a^4 b^4}{9 x^{18}}-\frac{7 a^3 b^5}{2 x^{16}}-\frac{2 a^2 b^6}{x^{14}}-\frac{a^7 b}{3 x^{24}}-\frac{a^8}{26 x^{26}}-\frac{2 a b^7}{3 x^{12}}-\frac{b^8}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^27,x]

[Out]

-a^8/(26*x^26) - (a^7*b)/(3*x^24) - (14*a^6*b^2)/(11*x^22) - (14*a^5*b^3)/(5*x^20) - (35*a^4*b^4)/(9*x^18) - (
7*a^3*b^5)/(2*x^16) - (2*a^2*b^6)/x^14 - (2*a*b^7)/(3*x^12) - b^8/(10*x^10)

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Maple [A]  time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{26\,{x}^{26}}}-{\frac{{a}^{7}b}{3\,{x}^{24}}}-{\frac{35\,{a}^{4}{b}^{4}}{9\,{x}^{18}}}-{\frac{{b}^{8}}{10\,{x}^{10}}}-{\frac{2\,a{b}^{7}}{3\,{x}^{12}}}-{\frac{7\,{a}^{3}{b}^{5}}{2\,{x}^{16}}}-{\frac{14\,{a}^{6}{b}^{2}}{11\,{x}^{22}}}-2\,{\frac{{a}^{2}{b}^{6}}{{x}^{14}}}-{\frac{14\,{a}^{5}{b}^{3}}{5\,{x}^{20}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^27,x)

[Out]

-1/26*a^8/x^26-1/3*a^7*b/x^24-35/9*a^4*b^4/x^18-1/10*b^8/x^10-2/3*a*b^7/x^12-7/2*a^3*b^5/x^16-14/11*a^6*b^2/x^
22-2*a^2*b^6/x^14-14/5*a^5*b^3/x^20

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Maxima [A]  time = 2.36697, size = 124, normalized size = 1.17 \begin{align*} -\frac{1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^27,x, algorithm="maxima")

[Out]

-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3*b^5*x^10 + 50050*a^4*b^4*x^8 + 3603
6*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 4290*a^7*b*x^2 + 495*a^8)/x^26

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Fricas [A]  time = 1.18707, size = 240, normalized size = 2.26 \begin{align*} -\frac{1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^27,x, algorithm="fricas")

[Out]

-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3*b^5*x^10 + 50050*a^4*b^4*x^8 + 3603
6*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 4290*a^7*b*x^2 + 495*a^8)/x^26

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Sympy [A]  time = 1.37786, size = 99, normalized size = 0.93 \begin{align*} - \frac{495 a^{8} + 4290 a^{7} b x^{2} + 16380 a^{6} b^{2} x^{4} + 36036 a^{5} b^{3} x^{6} + 50050 a^{4} b^{4} x^{8} + 45045 a^{3} b^{5} x^{10} + 25740 a^{2} b^{6} x^{12} + 8580 a b^{7} x^{14} + 1287 b^{8} x^{16}}{12870 x^{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**27,x)

[Out]

-(495*a**8 + 4290*a**7*b*x**2 + 16380*a**6*b**2*x**4 + 36036*a**5*b**3*x**6 + 50050*a**4*b**4*x**8 + 45045*a**
3*b**5*x**10 + 25740*a**2*b**6*x**12 + 8580*a*b**7*x**14 + 1287*b**8*x**16)/(12870*x**26)

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Giac [A]  time = 2.79553, size = 124, normalized size = 1.17 \begin{align*} -\frac{1287 \, b^{8} x^{16} + 8580 \, a b^{7} x^{14} + 25740 \, a^{2} b^{6} x^{12} + 45045 \, a^{3} b^{5} x^{10} + 50050 \, a^{4} b^{4} x^{8} + 36036 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 4290 \, a^{7} b x^{2} + 495 \, a^{8}}{12870 \, x^{26}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^27,x, algorithm="giac")

[Out]

-1/12870*(1287*b^8*x^16 + 8580*a*b^7*x^14 + 25740*a^2*b^6*x^12 + 45045*a^3*b^5*x^10 + 50050*a^4*b^4*x^8 + 3603
6*a^5*b^3*x^6 + 16380*a^6*b^2*x^4 + 4290*a^7*b*x^2 + 495*a^8)/x^26