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

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

[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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Rubi [A]  time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.00, size = 108, normalized size = 1.00 \[ -\frac {a^8}{28 x^{28}}-\frac {4 a^7 b}{13 x^{26}}-\frac {7 a^6 b^2}{6 x^{24}}-\frac {28 a^5 b^3}{11 x^{22}}-\frac {7 a^4 b^4}{2 x^{20}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {7 a^2 b^6}{4 x^{16}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^29,x]

[Out]

-1/28*a^8/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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fricas [A]  time = 0.88, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]

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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giac [A]  time = 1.05, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]

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

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maple [A]  time = 0.01, size = 91, normalized size = 0.84 \[ -\frac {b^{8}}{12 x^{12}}-\frac {4 a \,b^{7}}{7 x^{14}}-\frac {7 a^{2} b^{6}}{4 x^{16}}-\frac {28 a^{3} b^{5}}{9 x^{18}}-\frac {7 a^{4} b^{4}}{2 x^{20}}-\frac {28 a^{5} b^{3}}{11 x^{22}}-\frac {7 a^{6} b^{2}}{6 x^{24}}-\frac {4 a^{7} b}{13 x^{26}}-\frac {a^{8}}{28 x^{28}} \]

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 = 1.41, size = 92, normalized size = 0.85 \[ -\frac {3003 \, b^{8} x^{16} + 20592 \, a b^{7} x^{14} + 63063 \, a^{2} b^{6} x^{12} + 112112 \, a^{3} b^{5} x^{10} + 126126 \, a^{4} b^{4} x^{8} + 91728 \, a^{5} b^{3} x^{6} + 42042 \, a^{6} b^{2} x^{4} + 11088 \, a^{7} b x^{2} + 1287 \, a^{8}}{36036 \, x^{28}} \]

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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mupad [B]  time = 4.89, size = 92, normalized size = 0.85 \[ -\frac {\frac {a^8}{28}+\frac {4\,a^7\,b\,x^2}{13}+\frac {7\,a^6\,b^2\,x^4}{6}+\frac {28\,a^5\,b^3\,x^6}{11}+\frac {7\,a^4\,b^4\,x^8}{2}+\frac {28\,a^3\,b^5\,x^{10}}{9}+\frac {7\,a^2\,b^6\,x^{12}}{4}+\frac {4\,a\,b^7\,x^{14}}{7}+\frac {b^8\,x^{16}}{12}}{x^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^29,x)

[Out]

-(a^8/28 + (b^8*x^16)/12 + (4*a^7*b*x^2)/13 + (4*a*b^7*x^14)/7 + (7*a^6*b^2*x^4)/6 + (28*a^5*b^3*x^6)/11 + (7*
a^4*b^4*x^8)/2 + (28*a^3*b^5*x^10)/9 + (7*a^2*b^6*x^12)/4)/x^28

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sympy [A]  time = 1.20, size = 99, normalized size = 0.92 \[ \frac {- 1287 a^{8} - 11088 a^{7} b x^{2} - 42042 a^{6} b^{2} x^{4} - 91728 a^{5} b^{3} x^{6} - 126126 a^{4} b^{4} x^{8} - 112112 a^{3} b^{5} x^{10} - 63063 a^{2} b^{6} x^{12} - 20592 a b^{7} x^{14} - 3003 b^{8} x^{16}}{36036 x^{28}} \]

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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