3.106 \(\int \frac{(a+b x^2)^8}{x^{29}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]

[Out]

-a^8/(28*x^28) - (4*a^7*b)/(13*x^26) - (7*a^6*b^2)/(6*x^24) - (28*a^5*b^3)/(11*x^22) - (7*a^4*b^4)/(2*x^20) -
(28*a^3*b^5)/(9*x^18) - (7*a^2*b^6)/(4*x^16) - (4*a*b^7)/(7*x^14) - b^8/(12*x^12)

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Rubi [A]  time = 0.0536761, 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} \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^29,x]

[Out]

-a^8/(28*x^28) - (4*a^7*b)/(13*x^26) - (7*a^6*b^2)/(6*x^24) - (28*a^5*b^3)/(11*x^22) - (7*a^4*b^4)/(2*x^20) -
(28*a^3*b^5)/(9*x^18) - (7*a^2*b^6)/(4*x^16) - (4*a*b^7)/(7*x^14) - b^8/(12*x^12)

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^{29}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{15}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^{15}}+\frac{8 a^7 b}{x^{14}}+\frac{28 a^6 b^2}{x^{13}}+\frac{56 a^5 b^3}{x^{12}}+\frac{70 a^4 b^4}{x^{11}}+\frac{56 a^3 b^5}{x^{10}}+\frac{28 a^2 b^6}{x^9}+\frac{8 a b^7}{x^8}+\frac{b^8}{x^7}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{28 x^{28}}-\frac{4 a^7 b}{13 x^{26}}-\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0043779, size = 108, normalized size = 1. \[ -\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a^7 b}{13 x^{26}}-\frac{a^8}{28 x^{28}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^29,x]

[Out]

-a^8/(28*x^28) - (4*a^7*b)/(13*x^26) - (7*a^6*b^2)/(6*x^24) - (28*a^5*b^3)/(11*x^22) - (7*a^4*b^4)/(2*x^20) -
(28*a^3*b^5)/(9*x^18) - (7*a^2*b^6)/(4*x^16) - (4*a*b^7)/(7*x^14) - b^8/(12*x^12)

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Maple [A]  time = 0.007, size = 91, normalized size = 0.8 \begin{align*} -{\frac{{a}^{8}}{28\,{x}^{28}}}-{\frac{4\,{a}^{7}b}{13\,{x}^{26}}}-{\frac{7\,{a}^{6}{b}^{2}}{6\,{x}^{24}}}-{\frac{28\,{a}^{5}{b}^{3}}{11\,{x}^{22}}}-{\frac{7\,{a}^{4}{b}^{4}}{2\,{x}^{20}}}-{\frac{28\,{a}^{3}{b}^{5}}{9\,{x}^{18}}}-{\frac{7\,{a}^{2}{b}^{6}}{4\,{x}^{16}}}-{\frac{4\,a{b}^{7}}{7\,{x}^{14}}}-{\frac{{b}^{8}}{12\,{x}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^29,x)

[Out]

-1/28*a^8/x^28-4/13*a^7*b/x^26-7/6*a^6*b^2/x^24-28/11*a^5*b^3/x^22-7/2*a^4*b^4/x^20-28/9*a^3*b^5/x^18-7/4*a^2*
b^6/x^16-4/7*a*b^7/x^14-1/12*b^8/x^12

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Maxima [A]  time = 1.97239, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^29,x, algorithm="maxima")

[Out]

-1/36036*(3003*b^8*x^16 + 20592*a*b^7*x^14 + 63063*a^2*b^6*x^12 + 112112*a^3*b^5*x^10 + 126126*a^4*b^4*x^8 + 9
1728*a^5*b^3*x^6 + 42042*a^6*b^2*x^4 + 11088*a^7*b*x^2 + 1287*a^8)/x^28

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Fricas [A]  time = 1.29526, size = 247, normalized size = 2.29 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^29,x, algorithm="fricas")

[Out]

-1/36036*(3003*b^8*x^16 + 20592*a*b^7*x^14 + 63063*a^2*b^6*x^12 + 112112*a^3*b^5*x^10 + 126126*a^4*b^4*x^8 + 9
1728*a^5*b^3*x^6 + 42042*a^6*b^2*x^4 + 11088*a^7*b*x^2 + 1287*a^8)/x^28

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Sympy [A]  time = 1.49153, size = 99, normalized size = 0.92 \begin{align*} - \frac{1287 a^{8} + 11088 a^{7} b x^{2} + 42042 a^{6} b^{2} x^{4} + 91728 a^{5} b^{3} x^{6} + 126126 a^{4} b^{4} x^{8} + 112112 a^{3} b^{5} x^{10} + 63063 a^{2} b^{6} x^{12} + 20592 a b^{7} x^{14} + 3003 b^{8} x^{16}}{36036 x^{28}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**29,x)

[Out]

-(1287*a**8 + 11088*a**7*b*x**2 + 42042*a**6*b**2*x**4 + 91728*a**5*b**3*x**6 + 126126*a**4*b**4*x**8 + 112112
*a**3*b**5*x**10 + 63063*a**2*b**6*x**12 + 20592*a*b**7*x**14 + 3003*b**8*x**16)/(36036*x**28)

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Giac [A]  time = 2.08714, size = 124, normalized size = 1.15 \begin{align*} -\frac{3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^29,x, algorithm="giac")

[Out]

-1/36036*(3003*b^8*x^16 + 20592*a*b^7*x^14 + 63063*a^2*b^6*x^12 + 112112*a^3*b^5*x^10 + 126126*a^4*b^4*x^8 + 9
1728*a^5*b^3*x^6 + 42042*a^6*b^2*x^4 + 11088*a^7*b*x^2 + 1287*a^8)/x^28