3.120 \(\int \frac {(a+b x^2)^8}{x^{14}} \, dx\)

Optimal. Leaf size=98 \[ -\frac {a^8}{13 x^{13}}-\frac {8 a^7 b}{11 x^{11}}-\frac {28 a^6 b^2}{9 x^9}-\frac {8 a^5 b^3}{x^7}-\frac {14 a^4 b^4}{x^5}-\frac {56 a^3 b^5}{3 x^3}-\frac {28 a^2 b^6}{x}+8 a b^7 x+\frac {b^8 x^3}{3} \]

[Out]

-1/13*a^8/x^13-8/11*a^7*b/x^11-28/9*a^6*b^2/x^9-8*a^5*b^3/x^7-14*a^4*b^4/x^5-56/3*a^3*b^5/x^3-28*a^2*b^6/x+8*a
*b^7*x+1/3*b^8*x^3

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Rubi [A]  time = 0.04, antiderivative size = 98, 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} \[ -\frac {28 a^6 b^2}{9 x^9}-\frac {8 a^5 b^3}{x^7}-\frac {14 a^4 b^4}{x^5}-\frac {56 a^3 b^5}{3 x^3}-\frac {28 a^2 b^6}{x}-\frac {8 a^7 b}{11 x^{11}}-\frac {a^8}{13 x^{13}}+8 a b^7 x+\frac {b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^14,x]

[Out]

-a^8/(13*x^13) - (8*a^7*b)/(11*x^11) - (28*a^6*b^2)/(9*x^9) - (8*a^5*b^3)/x^7 - (14*a^4*b^4)/x^5 - (56*a^3*b^5
)/(3*x^3) - (28*a^2*b^6)/x + 8*a*b^7*x + (b^8*x^3)/3

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

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Mathematica [A]  time = 0.01, size = 98, normalized size = 1.00 \[ -\frac {a^8}{13 x^{13}}-\frac {8 a^7 b}{11 x^{11}}-\frac {28 a^6 b^2}{9 x^9}-\frac {8 a^5 b^3}{x^7}-\frac {14 a^4 b^4}{x^5}-\frac {56 a^3 b^5}{3 x^3}-\frac {28 a^2 b^6}{x}+8 a b^7 x+\frac {b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^14,x]

[Out]

-1/13*a^8/x^13 - (8*a^7*b)/(11*x^11) - (28*a^6*b^2)/(9*x^9) - (8*a^5*b^3)/x^7 - (14*a^4*b^4)/x^5 - (56*a^3*b^5
)/(3*x^3) - (28*a^2*b^6)/x + 8*a*b^7*x + (b^8*x^3)/3

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fricas [A]  time = 0.84, size = 92, normalized size = 0.94 \[ \frac {429 \, b^{8} x^{16} + 10296 \, a b^{7} x^{14} - 36036 \, a^{2} b^{6} x^{12} - 24024 \, a^{3} b^{5} x^{10} - 18018 \, a^{4} b^{4} x^{8} - 10296 \, a^{5} b^{3} x^{6} - 4004 \, a^{6} b^{2} x^{4} - 936 \, a^{7} b x^{2} - 99 \, a^{8}}{1287 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="fricas")

[Out]

1/1287*(429*b^8*x^16 + 10296*a*b^7*x^14 - 36036*a^2*b^6*x^12 - 24024*a^3*b^5*x^10 - 18018*a^4*b^4*x^8 - 10296*
a^5*b^3*x^6 - 4004*a^6*b^2*x^4 - 936*a^7*b*x^2 - 99*a^8)/x^13

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giac [A]  time = 1.10, size = 91, normalized size = 0.93 \[ \frac {1}{3} \, b^{8} x^{3} + 8 \, a b^{7} x - \frac {36036 \, a^{2} b^{6} x^{12} + 24024 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 10296 \, a^{5} b^{3} x^{6} + 4004 \, a^{6} b^{2} x^{4} + 936 \, a^{7} b x^{2} + 99 \, a^{8}}{1287 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="giac")

[Out]

1/3*b^8*x^3 + 8*a*b^7*x - 1/1287*(36036*a^2*b^6*x^12 + 24024*a^3*b^5*x^10 + 18018*a^4*b^4*x^8 + 10296*a^5*b^3*
x^6 + 4004*a^6*b^2*x^4 + 936*a^7*b*x^2 + 99*a^8)/x^13

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maple [A]  time = 0.01, size = 89, normalized size = 0.91 \[ \frac {b^{8} x^{3}}{3}+8 a \,b^{7} x -\frac {28 a^{2} b^{6}}{x}-\frac {56 a^{3} b^{5}}{3 x^{3}}-\frac {14 a^{4} b^{4}}{x^{5}}-\frac {8 a^{5} b^{3}}{x^{7}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {8 a^{7} b}{11 x^{11}}-\frac {a^{8}}{13 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^14,x)

[Out]

-1/13*a^8/x^13-8/11*a^7*b/x^11-28/9*a^6*b^2/x^9-8*a^5*b^3/x^7-14*a^4*b^4/x^5-56/3*a^3*b^5/x^3-28*a^2*b^6/x+8*a
*b^7*x+1/3*b^8*x^3

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maxima [A]  time = 1.33, size = 91, normalized size = 0.93 \[ \frac {1}{3} \, b^{8} x^{3} + 8 \, a b^{7} x - \frac {36036 \, a^{2} b^{6} x^{12} + 24024 \, a^{3} b^{5} x^{10} + 18018 \, a^{4} b^{4} x^{8} + 10296 \, a^{5} b^{3} x^{6} + 4004 \, a^{6} b^{2} x^{4} + 936 \, a^{7} b x^{2} + 99 \, a^{8}}{1287 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^14,x, algorithm="maxima")

[Out]

1/3*b^8*x^3 + 8*a*b^7*x - 1/1287*(36036*a^2*b^6*x^12 + 24024*a^3*b^5*x^10 + 18018*a^4*b^4*x^8 + 10296*a^5*b^3*
x^6 + 4004*a^6*b^2*x^4 + 936*a^7*b*x^2 + 99*a^8)/x^13

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mupad [B]  time = 0.07, size = 92, normalized size = 0.94 \[ -\frac {\frac {a^8}{13}+\frac {8\,a^7\,b\,x^2}{11}+\frac {28\,a^6\,b^2\,x^4}{9}+8\,a^5\,b^3\,x^6+14\,a^4\,b^4\,x^8+\frac {56\,a^3\,b^5\,x^{10}}{3}+28\,a^2\,b^6\,x^{12}-8\,a\,b^7\,x^{14}-\frac {b^8\,x^{16}}{3}}{x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^14,x)

[Out]

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

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sympy [A]  time = 0.57, size = 97, normalized size = 0.99 \[ 8 a b^{7} x + \frac {b^{8} x^{3}}{3} + \frac {- 99 a^{8} - 936 a^{7} b x^{2} - 4004 a^{6} b^{2} x^{4} - 10296 a^{5} b^{3} x^{6} - 18018 a^{4} b^{4} x^{8} - 24024 a^{3} b^{5} x^{10} - 36036 a^{2} b^{6} x^{12}}{1287 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**14,x)

[Out]

8*a*b**7*x + b**8*x**3/3 + (-99*a**8 - 936*a**7*b*x**2 - 4004*a**6*b**2*x**4 - 10296*a**5*b**3*x**6 - 18018*a*
*4*b**4*x**8 - 24024*a**3*b**5*x**10 - 36036*a**2*b**6*x**12)/(1287*x**13)

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