3.122 \(\int \frac {(a+b x^2)^8}{x^{18}} \, dx\)

Optimal. Leaf size=104 \[ -\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x} \]

[Out]

-1/17*a^8/x^17-8/15*a^7*b/x^15-28/13*a^6*b^2/x^13-56/11*a^5*b^3/x^11-70/9*a^4*b^4/x^9-8*a^3*b^5/x^7-28/5*a^2*b
^6/x^5-8/3*a*b^7/x^3-b^8/x

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ -\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a^7 b}{15 x^{15}}-\frac {a^8}{17 x^{17}}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^18,x]

[Out]

-a^8/(17*x^17) - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^3)/(11*x^11) - (70*a^4*b^4)/(9*x^9)
- (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^5) - (8*a*b^7)/(3*x^3) - b^8/x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx &=\int \left (\frac {a^8}{x^{18}}+\frac {8 a^7 b}{x^{16}}+\frac {28 a^6 b^2}{x^{14}}+\frac {56 a^5 b^3}{x^{12}}+\frac {70 a^4 b^4}{x^{10}}+\frac {56 a^3 b^5}{x^8}+\frac {28 a^2 b^6}{x^6}+\frac {8 a b^7}{x^4}+\frac {b^8}{x^2}\right ) \, dx\\ &=-\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 104, normalized size = 1.00 \[ -\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^18,x]

[Out]

-1/17*a^8/x^17 - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^3)/(11*x^11) - (70*a^4*b^4)/(9*x^9)
- (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^5) - (8*a*b^7)/(3*x^3) - b^8/x

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 92, normalized size = 0.88 \[ -\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^18,x, algorithm="fricas")

[Out]

-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875160*a^3*b^5*x^10 + 850850*a^4*b^4*x^
8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17

________________________________________________________________________________________

giac [A]  time = 0.82, size = 92, normalized size = 0.88 \[ -\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^18,x, algorithm="giac")

[Out]

-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875160*a^3*b^5*x^10 + 850850*a^4*b^4*x^
8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17

________________________________________________________________________________________

maple [A]  time = 0.01, size = 91, normalized size = 0.88 \[ -\frac {b^{8}}{x}-\frac {8 a \,b^{7}}{3 x^{3}}-\frac {28 a^{2} b^{6}}{5 x^{5}}-\frac {8 a^{3} b^{5}}{x^{7}}-\frac {70 a^{4} b^{4}}{9 x^{9}}-\frac {56 a^{5} b^{3}}{11 x^{11}}-\frac {28 a^{6} b^{2}}{13 x^{13}}-\frac {8 a^{7} b}{15 x^{15}}-\frac {a^{8}}{17 x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^18,x)

[Out]

-1/17*a^8/x^17-8/15*a^7*b/x^15-28/13*a^6*b^2/x^13-56/11*a^5*b^3/x^11-70/9*a^4*b^4/x^9-8*a^3*b^5/x^7-28/5*a^2*b
^6/x^5-8/3*a*b^7/x^3-b^8/x

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 92, normalized size = 0.88 \[ -\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^18,x, algorithm="maxima")

[Out]

-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875160*a^3*b^5*x^10 + 850850*a^4*b^4*x^
8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 91, normalized size = 0.88 \[ -\frac {\frac {a^8}{17}+\frac {8\,a^7\,b\,x^2}{15}+\frac {28\,a^6\,b^2\,x^4}{13}+\frac {56\,a^5\,b^3\,x^6}{11}+\frac {70\,a^4\,b^4\,x^8}{9}+8\,a^3\,b^5\,x^{10}+\frac {28\,a^2\,b^6\,x^{12}}{5}+\frac {8\,a\,b^7\,x^{14}}{3}+b^8\,x^{16}}{x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^18,x)

[Out]

-(a^8/17 + b^8*x^16 + (8*a^7*b*x^2)/15 + (8*a*b^7*x^14)/3 + (28*a^6*b^2*x^4)/13 + (56*a^5*b^3*x^6)/11 + (70*a^
4*b^4*x^8)/9 + 8*a^3*b^5*x^10 + (28*a^2*b^6*x^12)/5)/x^17

________________________________________________________________________________________

sympy [A]  time = 0.79, size = 99, normalized size = 0.95 \[ \frac {- 6435 a^{8} - 58344 a^{7} b x^{2} - 235620 a^{6} b^{2} x^{4} - 556920 a^{5} b^{3} x^{6} - 850850 a^{4} b^{4} x^{8} - 875160 a^{3} b^{5} x^{10} - 612612 a^{2} b^{6} x^{12} - 291720 a b^{7} x^{14} - 109395 b^{8} x^{16}}{109395 x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**18,x)

[Out]

(-6435*a**8 - 58344*a**7*b*x**2 - 235620*a**6*b**2*x**4 - 556920*a**5*b**3*x**6 - 850850*a**4*b**4*x**8 - 8751
60*a**3*b**5*x**10 - 612612*a**2*b**6*x**12 - 291720*a*b**7*x**14 - 109395*b**8*x**16)/(109395*x**17)

________________________________________________________________________________________