3.108 \(\int \frac{(a+b x^2)^8}{x^{33}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{32 x^{32}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) - (35*a^4*b^4)/(12*x^24) - (28*
a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)

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Rubi [A]  time = 0.0517498, antiderivative size = 106, 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} \[ -\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{32 x^{32}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^33,x]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) - (35*a^4*b^4)/(12*x^24) - (28*
a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)

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^{33}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{17}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^{17}}+\frac{8 a^7 b}{x^{16}}+\frac{28 a^6 b^2}{x^{15}}+\frac{56 a^5 b^3}{x^{14}}+\frac{70 a^4 b^4}{x^{13}}+\frac{56 a^3 b^5}{x^{12}}+\frac{28 a^2 b^6}{x^{11}}+\frac{8 a b^7}{x^{10}}+\frac{b^8}{x^9}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}}\\ \end{align*}

Mathematica [A]  time = 0.0040941, size = 106, normalized size = 1. \[ -\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{32 x^{32}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^33,x]

[Out]

-a^8/(32*x^32) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) - (35*a^4*b^4)/(12*x^24) - (28*
a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)

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Maple [A]  time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{32\,{x}^{32}}}-{\frac{4\,{a}^{7}b}{15\,{x}^{30}}}-{\frac{{a}^{6}{b}^{2}}{{x}^{28}}}-{\frac{28\,{a}^{5}{b}^{3}}{13\,{x}^{26}}}-{\frac{35\,{a}^{4}{b}^{4}}{12\,{x}^{24}}}-{\frac{28\,{a}^{3}{b}^{5}}{11\,{x}^{22}}}-{\frac{7\,{a}^{2}{b}^{6}}{5\,{x}^{20}}}-{\frac{4\,a{b}^{7}}{9\,{x}^{18}}}-{\frac{{b}^{8}}{16\,{x}^{16}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^33,x)

[Out]

-1/32*a^8/x^32-4/15*a^7*b/x^30-a^6*b^2/x^28-28/13*a^5*b^3/x^26-35/12*a^4*b^4/x^24-28/11*a^3*b^5/x^22-7/5*a^2*b
^6/x^20-4/9*a*b^7/x^18-1/16*b^8/x^16

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Maxima [A]  time = 1.34983, size = 124, normalized size = 1.17 \begin{align*} -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^33,x, algorithm="maxima")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*b^5*x^10 + 600600*a^4*b^4*x^8
+ 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32

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Fricas [A]  time = 1.25686, size = 254, normalized size = 2.4 \begin{align*} -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^33,x, algorithm="fricas")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*b^5*x^10 + 600600*a^4*b^4*x^8
+ 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32

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Sympy [A]  time = 1.83298, size = 99, normalized size = 0.93 \begin{align*} - \frac{6435 a^{8} + 54912 a^{7} b x^{2} + 205920 a^{6} b^{2} x^{4} + 443520 a^{5} b^{3} x^{6} + 600600 a^{4} b^{4} x^{8} + 524160 a^{3} b^{5} x^{10} + 288288 a^{2} b^{6} x^{12} + 91520 a b^{7} x^{14} + 12870 b^{8} x^{16}}{205920 x^{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**33,x)

[Out]

-(6435*a**8 + 54912*a**7*b*x**2 + 205920*a**6*b**2*x**4 + 443520*a**5*b**3*x**6 + 600600*a**4*b**4*x**8 + 5241
60*a**3*b**5*x**10 + 288288*a**2*b**6*x**12 + 91520*a*b**7*x**14 + 12870*b**8*x**16)/(205920*x**32)

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Giac [A]  time = 1.86592, size = 124, normalized size = 1.17 \begin{align*} -\frac{12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^33,x, algorithm="giac")

[Out]

-1/205920*(12870*b^8*x^16 + 91520*a*b^7*x^14 + 288288*a^2*b^6*x^12 + 524160*a^3*b^5*x^10 + 600600*a^4*b^4*x^8
+ 443520*a^5*b^3*x^6 + 205920*a^6*b^2*x^4 + 54912*a^7*b*x^2 + 6435*a^8)/x^32