3.107 \(\int \frac{(a+b x^2)^8}{x^{31}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{2 a^7 b}{7 x^{28}}-\frac{a^8}{30 x^{30}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}} \]

[Out]

-a^8/(30*x^30) - (2*a^7*b)/(7*x^28) - (14*a^6*b^2)/(13*x^26) - (7*a^5*b^3)/(3*x^24) - (35*a^4*b^4)/(11*x^22) -
(14*a^3*b^5)/(5*x^20) - (14*a^2*b^6)/(9*x^18) - (a*b^7)/(2*x^16) - b^8/(14*x^14)

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Rubi [A]  time = 0.0522581, antiderivative size = 108, 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} \[ -\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{2 a^7 b}{7 x^{28}}-\frac{a^8}{30 x^{30}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^31,x]

[Out]

-a^8/(30*x^30) - (2*a^7*b)/(7*x^28) - (14*a^6*b^2)/(13*x^26) - (7*a^5*b^3)/(3*x^24) - (35*a^4*b^4)/(11*x^22) -
(14*a^3*b^5)/(5*x^20) - (14*a^2*b^6)/(9*x^18) - (a*b^7)/(2*x^16) - b^8/(14*x^14)

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^{31}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{16}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^{16}}+\frac{8 a^7 b}{x^{15}}+\frac{28 a^6 b^2}{x^{14}}+\frac{56 a^5 b^3}{x^{13}}+\frac{70 a^4 b^4}{x^{12}}+\frac{56 a^3 b^5}{x^{11}}+\frac{28 a^2 b^6}{x^{10}}+\frac{8 a b^7}{x^9}+\frac{b^8}{x^8}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{30 x^{30}}-\frac{2 a^7 b}{7 x^{28}}-\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}}\\ \end{align*}

Mathematica [A]  time = 0.004014, size = 108, normalized size = 1. \[ -\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{2 a^7 b}{7 x^{28}}-\frac{a^8}{30 x^{30}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^31,x]

[Out]

-a^8/(30*x^30) - (2*a^7*b)/(7*x^28) - (14*a^6*b^2)/(13*x^26) - (7*a^5*b^3)/(3*x^24) - (35*a^4*b^4)/(11*x^22) -
(14*a^3*b^5)/(5*x^20) - (14*a^2*b^6)/(9*x^18) - (a*b^7)/(2*x^16) - b^8/(14*x^14)

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Maple [A]  time = 0.008, size = 91, normalized size = 0.8 \begin{align*} -{\frac{{a}^{8}}{30\,{x}^{30}}}-{\frac{2\,{a}^{7}b}{7\,{x}^{28}}}-{\frac{14\,{a}^{6}{b}^{2}}{13\,{x}^{26}}}-{\frac{7\,{a}^{5}{b}^{3}}{3\,{x}^{24}}}-{\frac{35\,{a}^{4}{b}^{4}}{11\,{x}^{22}}}-{\frac{14\,{a}^{3}{b}^{5}}{5\,{x}^{20}}}-{\frac{14\,{a}^{2}{b}^{6}}{9\,{x}^{18}}}-{\frac{a{b}^{7}}{2\,{x}^{16}}}-{\frac{{b}^{8}}{14\,{x}^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^31,x)

[Out]

-1/30*a^8/x^30-2/7*a^7*b/x^28-14/13*a^6*b^2/x^26-7/3*a^5*b^3/x^24-35/11*a^4*b^4/x^22-14/5*a^3*b^5/x^20-14/9*a^
2*b^6/x^18-1/2*a*b^7/x^16-1/14*b^8/x^14

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Maxima [A]  time = 1.99641, size = 124, normalized size = 1.15 \begin{align*} -\frac{6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^31,x, algorithm="maxima")

[Out]

-1/90090*(6435*b^8*x^16 + 45045*a*b^7*x^14 + 140140*a^2*b^6*x^12 + 252252*a^3*b^5*x^10 + 286650*a^4*b^4*x^8 +
210210*a^5*b^3*x^6 + 97020*a^6*b^2*x^4 + 25740*a^7*b*x^2 + 3003*a^8)/x^30

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Fricas [A]  time = 1.1025, size = 250, normalized size = 2.31 \begin{align*} -\frac{6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^31,x, algorithm="fricas")

[Out]

-1/90090*(6435*b^8*x^16 + 45045*a*b^7*x^14 + 140140*a^2*b^6*x^12 + 252252*a^3*b^5*x^10 + 286650*a^4*b^4*x^8 +
210210*a^5*b^3*x^6 + 97020*a^6*b^2*x^4 + 25740*a^7*b*x^2 + 3003*a^8)/x^30

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Sympy [A]  time = 1.6282, size = 99, normalized size = 0.92 \begin{align*} - \frac{3003 a^{8} + 25740 a^{7} b x^{2} + 97020 a^{6} b^{2} x^{4} + 210210 a^{5} b^{3} x^{6} + 286650 a^{4} b^{4} x^{8} + 252252 a^{3} b^{5} x^{10} + 140140 a^{2} b^{6} x^{12} + 45045 a b^{7} x^{14} + 6435 b^{8} x^{16}}{90090 x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**31,x)

[Out]

-(3003*a**8 + 25740*a**7*b*x**2 + 97020*a**6*b**2*x**4 + 210210*a**5*b**3*x**6 + 286650*a**4*b**4*x**8 + 25225
2*a**3*b**5*x**10 + 140140*a**2*b**6*x**12 + 45045*a*b**7*x**14 + 6435*b**8*x**16)/(90090*x**30)

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Giac [A]  time = 2.98414, size = 124, normalized size = 1.15 \begin{align*} -\frac{6435 \, b^{8} x^{16} + 45045 \, a b^{7} x^{14} + 140140 \, a^{2} b^{6} x^{12} + 252252 \, a^{3} b^{5} x^{10} + 286650 \, a^{4} b^{4} x^{8} + 210210 \, a^{5} b^{3} x^{6} + 97020 \, a^{6} b^{2} x^{4} + 25740 \, a^{7} b x^{2} + 3003 \, a^{8}}{90090 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^31,x, algorithm="giac")

[Out]

-1/90090*(6435*b^8*x^16 + 45045*a*b^7*x^14 + 140140*a^2*b^6*x^12 + 252252*a^3*b^5*x^10 + 286650*a^4*b^4*x^8 +
210210*a^5*b^3*x^6 + 97020*a^6*b^2*x^4 + 25740*a^7*b*x^2 + 3003*a^8)/x^30