3.99 \(\int \frac {(a+b x^2)^8}{x^{15}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]

[Out]

-1/14*a^8/x^14-2/3*a^7*b/x^12-14/5*a^6*b^2/x^10-7*a^5*b^3/x^8-35/3*a^4*b^4/x^6-14*a^3*b^5/x^4-14*a^2*b^6/x^2+1
/2*b^8*x^2+8*a*b^7*ln(x)

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Rubi [A]  time = 0.05, antiderivative size = 99, 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} \[ -\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}-\frac {2 a^7 b}{3 x^{12}}-\frac {a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^15,x]

[Out]

-a^8/(14*x^14) - (2*a^7*b)/(3*x^12) - (14*a^6*b^2)/(5*x^10) - (7*a^5*b^3)/x^8 - (35*a^4*b^4)/(3*x^6) - (14*a^3
*b^5)/x^4 - (14*a^2*b^6)/x^2 + (b^8*x^2)/2 + 8*a*b^7*Log[x]

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^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^8} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^8+\frac {a^8}{x^8}+\frac {8 a^7 b}{x^7}+\frac {28 a^6 b^2}{x^6}+\frac {56 a^5 b^3}{x^5}+\frac {70 a^4 b^4}{x^4}+\frac {56 a^3 b^5}{x^3}+\frac {28 a^2 b^6}{x^2}+\frac {8 a b^7}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+\frac {b^8 x^2}{2}+8 a b^7 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 99, normalized size = 1.00 \[ -\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^15,x]

[Out]

-1/14*a^8/x^14 - (2*a^7*b)/(3*x^12) - (14*a^6*b^2)/(5*x^10) - (7*a^5*b^3)/x^8 - (35*a^4*b^4)/(3*x^6) - (14*a^3
*b^5)/x^4 - (14*a^2*b^6)/x^2 + (b^8*x^2)/2 + 8*a*b^7*Log[x]

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fricas [A]  time = 0.73, size = 94, normalized size = 0.95 \[ \frac {105 \, b^{8} x^{16} + 1680 \, a b^{7} x^{14} \log \relax (x) - 2940 \, a^{2} b^{6} x^{12} - 2940 \, a^{3} b^{5} x^{10} - 2450 \, a^{4} b^{4} x^{8} - 1470 \, a^{5} b^{3} x^{6} - 588 \, a^{6} b^{2} x^{4} - 140 \, a^{7} b x^{2} - 15 \, a^{8}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^15,x, algorithm="fricas")

[Out]

1/210*(105*b^8*x^16 + 1680*a*b^7*x^14*log(x) - 2940*a^2*b^6*x^12 - 2940*a^3*b^5*x^10 - 2450*a^4*b^4*x^8 - 1470
*a^5*b^3*x^6 - 588*a^6*b^2*x^4 - 140*a^7*b*x^2 - 15*a^8)/x^14

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giac [A]  time = 1.08, size = 103, normalized size = 1.04 \[ \frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2178 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^15,x, algorithm="giac")

[Out]

1/2*b^8*x^2 + 4*a*b^7*log(x^2) - 1/210*(2178*a*b^7*x^14 + 2940*a^2*b^6*x^12 + 2940*a^3*b^5*x^10 + 2450*a^4*b^4
*x^8 + 1470*a^5*b^3*x^6 + 588*a^6*b^2*x^4 + 140*a^7*b*x^2 + 15*a^8)/x^14

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maple [A]  time = 0.01, size = 90, normalized size = 0.91 \[ \frac {b^{8} x^{2}}{2}+8 a \,b^{7} \ln \relax (x )-\frac {14 a^{2} b^{6}}{x^{2}}-\frac {14 a^{3} b^{5}}{x^{4}}-\frac {35 a^{4} b^{4}}{3 x^{6}}-\frac {7 a^{5} b^{3}}{x^{8}}-\frac {14 a^{6} b^{2}}{5 x^{10}}-\frac {2 a^{7} b}{3 x^{12}}-\frac {a^{8}}{14 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^15,x)

[Out]

-1/14*a^8/x^14-2/3*a^7*b/x^12-14/5*a^6*b^2/x^10-7*a^5*b^3/x^8-35/3*a^4*b^4/x^6-14*a^3*b^5/x^4-14*a^2*b^6/x^2+1
/2*b^8*x^2+8*a*b^7*ln(x)

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maxima [A]  time = 1.35, size = 94, normalized size = 0.95 \[ \frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^15,x, algorithm="maxima")

[Out]

1/2*b^8*x^2 + 4*a*b^7*log(x^2) - 1/210*(2940*a^2*b^6*x^12 + 2940*a^3*b^5*x^10 + 2450*a^4*b^4*x^8 + 1470*a^5*b^
3*x^6 + 588*a^6*b^2*x^4 + 140*a^7*b*x^2 + 15*a^8)/x^14

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mupad [B]  time = 5.15, size = 94, normalized size = 0.95 \[ -\frac {\frac {a^8}{14}-\frac {b^8\,x^{16}}{2}+\frac {2\,a^7\,b\,x^2}{3}+\frac {14\,a^6\,b^2\,x^4}{5}+7\,a^5\,b^3\,x^6+\frac {35\,a^4\,b^4\,x^8}{3}+14\,a^3\,b^5\,x^{10}+14\,a^2\,b^6\,x^{12}-8\,a\,b^7\,x^{14}\,\ln \relax (x)}{x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^15,x)

[Out]

-(a^8/14 - (b^8*x^16)/2 + (2*a^7*b*x^2)/3 + (14*a^6*b^2*x^4)/5 + 7*a^5*b^3*x^6 + (35*a^4*b^4*x^8)/3 + 14*a^3*b
^5*x^10 + 14*a^2*b^6*x^12 - 8*a*b^7*x^14*log(x))/x^14

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sympy [A]  time = 0.73, size = 99, normalized size = 1.00 \[ 8 a b^{7} \log {\relax (x )} + \frac {b^{8} x^{2}}{2} + \frac {- 15 a^{8} - 140 a^{7} b x^{2} - 588 a^{6} b^{2} x^{4} - 1470 a^{5} b^{3} x^{6} - 2450 a^{4} b^{4} x^{8} - 2940 a^{3} b^{5} x^{10} - 2940 a^{2} b^{6} x^{12}}{210 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**15,x)

[Out]

8*a*b**7*log(x) + b**8*x**2/2 + (-15*a**8 - 140*a**7*b*x**2 - 588*a**6*b**2*x**4 - 1470*a**5*b**3*x**6 - 2450*
a**4*b**4*x**8 - 2940*a**3*b**5*x**10 - 2940*a**2*b**6*x**12)/(210*x**14)

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