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

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

[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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Rubi [A]  time = 0.05, antiderivative size = 106, 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} \[ -\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^8}{32 x^{32}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \]

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 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^{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*}

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Mathematica [A]  time = 0.00, size = 106, normalized size = 1.00 \[ -\frac {a^8}{32 x^{32}}-\frac {4 a^7 b}{15 x^{30}}-\frac {a^6 b^2}{x^{28}}-\frac {28 a^5 b^3}{13 x^{26}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {28 a^3 b^5}{11 x^{22}}-\frac {7 a^2 b^6}{5 x^{20}}-\frac {4 a b^7}{9 x^{18}}-\frac {b^8}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^33,x]

[Out]

-1/32*a^8/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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fricas [A]  time = 0.60, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

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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giac [A]  time = 1.05, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

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

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maple [A]  time = 0.01, size = 91, normalized size = 0.86 \[ -\frac {b^{8}}{16 x^{16}}-\frac {4 a \,b^{7}}{9 x^{18}}-\frac {7 a^{2} b^{6}}{5 x^{20}}-\frac {28 a^{3} b^{5}}{11 x^{22}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {28 a^{5} b^{3}}{13 x^{26}}-\frac {a^{6} b^{2}}{x^{28}}-\frac {4 a^{7} b}{15 x^{30}}-\frac {a^{8}}{32 x^{32}} \]

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 = 1.34, size = 92, normalized size = 0.87 \[ -\frac {12870 \, b^{8} x^{16} + 91520 \, a b^{7} x^{14} + 288288 \, a^{2} b^{6} x^{12} + 524160 \, a^{3} b^{5} x^{10} + 600600 \, a^{4} b^{4} x^{8} + 443520 \, a^{5} b^{3} x^{6} + 205920 \, a^{6} b^{2} x^{4} + 54912 \, a^{7} b x^{2} + 6435 \, a^{8}}{205920 \, x^{32}} \]

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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mupad [B]  time = 0.08, size = 91, normalized size = 0.86 \[ -\frac {\frac {a^8}{32}+\frac {4\,a^7\,b\,x^2}{15}+a^6\,b^2\,x^4+\frac {28\,a^5\,b^3\,x^6}{13}+\frac {35\,a^4\,b^4\,x^8}{12}+\frac {28\,a^3\,b^5\,x^{10}}{11}+\frac {7\,a^2\,b^6\,x^{12}}{5}+\frac {4\,a\,b^7\,x^{14}}{9}+\frac {b^8\,x^{16}}{16}}{x^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^33,x)

[Out]

-(a^8/32 + (b^8*x^16)/16 + (4*a^7*b*x^2)/15 + (4*a*b^7*x^14)/9 + a^6*b^2*x^4 + (28*a^5*b^3*x^6)/13 + (35*a^4*b
^4*x^8)/12 + (28*a^3*b^5*x^10)/11 + (7*a^2*b^6*x^12)/5)/x^32

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sympy [A]  time = 1.34, size = 99, normalized size = 0.93 \[ \frac {- 6435 a^{8} - 54912 a^{7} b x^{2} - 205920 a^{6} b^{2} x^{4} - 443520 a^{5} b^{3} x^{6} - 600600 a^{4} b^{4} x^{8} - 524160 a^{3} b^{5} x^{10} - 288288 a^{2} b^{6} x^{12} - 91520 a b^{7} x^{14} - 12870 b^{8} x^{16}}{205920 x^{32}} \]

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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