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

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

[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

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Rubi [A]  time = 0.0393411, antiderivative size = 104, normalized size of antiderivative = 1., 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 = 0.0097595, size = 104, normalized size = 1. \[ -\frac{28 a^6 b^2}{13 x^{13}}-\frac{56 a^5 b^3}{11 x^{11}}-\frac{70 a^4 b^4}{9 x^9}-\frac{8 a^3 b^5}{x^7}-\frac{28 a^2 b^6}{5 x^5}-\frac{8 a^7 b}{15 x^{15}}-\frac{a^8}{17 x^{17}}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(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

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Maple [A]  time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\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}}}-8\,{\frac{{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*}

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

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Maxima [A]  time = 1.54452, size = 124, normalized size = 1.19 \begin{align*} -\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}} \end{align*}

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

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Fricas [A]  time = 1.17504, size = 257, normalized size = 2.47 \begin{align*} -\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}} \end{align*}

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

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Sympy [A]  time = 1.00391, size = 99, normalized size = 0.95 \begin{align*} - \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}} \end{align*}

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)

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Giac [A]  time = 2.39408, size = 124, normalized size = 1.19 \begin{align*} -\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}} \end{align*}

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