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

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

[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]

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Rubi [A]  time = 0.0514795, antiderivative size = 99, 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} \[ -\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac{b^8 x^2}{2} \]

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 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^{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*}

Mathematica [A]  time = 0.0045511, size = 99, normalized size = 1. \[ -\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac{b^8 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{14\,{x}^{14}}}-{\frac{2\,{a}^{7}b}{3\,{x}^{12}}}-{\frac{14\,{a}^{6}{b}^{2}}{5\,{x}^{10}}}-7\,{\frac{{a}^{5}{b}^{3}}{{x}^{8}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-14\,{\frac{{a}^{3}{b}^{5}}{{x}^{4}}}-14\,{\frac{{a}^{2}{b}^{6}}{{x}^{2}}}+{\frac{{b}^{8}{x}^{2}}{2}}+8\,a{b}^{7}\ln \left ( x \right ) \end{align*}

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 = 2.33088, size = 127, normalized size = 1.28 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.27335, size = 234, normalized size = 2.36 \begin{align*} \frac{105 \, b^{8} x^{16} + 1680 \, a b^{7} x^{14} \log \left (x\right ) - 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}} \end{align*}

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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Sympy [A]  time = 0.955497, size = 97, normalized size = 0.98 \begin{align*} 8 a b^{7} \log{\left (x \right )} + \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}} \end{align*}

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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Giac [A]  time = 2.27729, size = 139, normalized size = 1.4 \begin{align*} \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}} \end{align*}

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