3.95 \(\int \frac{(a+b x^2)^8}{x^7} \, dx\)

Optimal. Leaf size=94 \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

[Out]

-a^8/(6*x^6) - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7
*x^8 + (b^8*x^10)/10 + 56*a^5*b^3*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0570915, antiderivative size = 94, 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} \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0046497, size = 94, normalized size = 1. \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7
*x^8 + (b^8*x^10)/10 + 56*a^5*b^3*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 89, normalized size = 1. \begin{align*} -{\frac{{a}^{8}}{6\,{x}^{6}}}-2\,{\frac{{a}^{7}b}{{x}^{4}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}+35\,{a}^{4}{b}^{4}{x}^{2}+14\,{a}^{3}{b}^{5}{x}^{4}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+a{b}^{7}{x}^{8}+{\frac{{b}^{8}{x}^{10}}{10}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^7,x)

[Out]

-1/6*a^8/x^6-2*a^7*b/x^4-14*a^6*b^2/x^2+35*a^4*b^4*x^2+14*a^3*b^5*x^4+14/3*a^2*b^6*x^6+a*b^7*x^8+1/10*b^8*x^10
+56*a^5*b^3*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 1.49857, size = 123, normalized size = 1.31 \begin{align*} \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac{84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="maxima")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 + 28*a^5*b^3*log(x^2) - 1/6*(84
*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8)/x^6

________________________________________________________________________________________

Fricas [A]  time = 1.36912, size = 220, normalized size = 2.34 \begin{align*} \frac{3 \, b^{8} x^{16} + 30 \, a b^{7} x^{14} + 140 \, a^{2} b^{6} x^{12} + 420 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{4} - 60 \, a^{7} b x^{2} - 5 \, a^{8}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="fricas")

[Out]

1/30*(3*b^8*x^16 + 30*a*b^7*x^14 + 140*a^2*b^6*x^12 + 420*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 1680*a^5*b^3*x^6*l
og(x) - 420*a^6*b^2*x^4 - 60*a^7*b*x^2 - 5*a^8)/x^6

________________________________________________________________________________________

Sympy [A]  time = 0.501642, size = 95, normalized size = 1.01 \begin{align*} 56 a^{5} b^{3} \log{\left (x \right )} + 35 a^{4} b^{4} x^{2} + 14 a^{3} b^{5} x^{4} + \frac{14 a^{2} b^{6} x^{6}}{3} + a b^{7} x^{8} + \frac{b^{8} x^{10}}{10} - \frac{a^{8} + 12 a^{7} b x^{2} + 84 a^{6} b^{2} x^{4}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**7,x)

[Out]

56*a**5*b**3*log(x) + 35*a**4*b**4*x**2 + 14*a**3*b**5*x**4 + 14*a**2*b**6*x**6/3 + a*b**7*x**8 + b**8*x**10/1
0 - (a**8 + 12*a**7*b*x**2 + 84*a**6*b**2*x**4)/(6*x**6)

________________________________________________________________________________________

Giac [A]  time = 3.88237, size = 138, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac{308 \, a^{5} b^{3} x^{6} + 84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="giac")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 + 28*a^5*b^3*log(x^2) - 1/6*(30
8*a^5*b^3*x^6 + 84*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8)/x^6