3.94 \(\int \frac{(a+b x^2)^8}{x^5} \, dx\)

Optimal. Leaf size=101 \[ \frac{7}{2} a^2 b^6 x^8+\frac{28}{3} a^3 b^5 x^6+\frac{35}{2} a^4 b^4 x^4+28 a^5 b^3 x^2+28 a^6 b^2 \log (x)-\frac{4 a^7 b}{x^2}-\frac{a^8}{4 x^4}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^{12}}{12} \]

[Out]

-a^8/(4*x^4) - (4*a^7*b)/x^2 + 28*a^5*b^3*x^2 + (35*a^4*b^4*x^4)/2 + (28*a^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 +
(4*a*b^7*x^10)/5 + (b^8*x^12)/12 + 28*a^6*b^2*Log[x]

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Rubi [A]  time = 0.061354, antiderivative size = 101, 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} \[ \frac{7}{2} a^2 b^6 x^8+\frac{28}{3} a^3 b^5 x^6+\frac{35}{2} a^4 b^4 x^4+28 a^5 b^3 x^2+28 a^6 b^2 \log (x)-\frac{4 a^7 b}{x^2}-\frac{a^8}{4 x^4}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0044516, size = 101, normalized size = 1. \[ \frac{7}{2} a^2 b^6 x^8+\frac{28}{3} a^3 b^5 x^6+\frac{35}{2} a^4 b^4 x^4+28 a^5 b^3 x^2+28 a^6 b^2 \log (x)-\frac{4 a^7 b}{x^2}-\frac{a^8}{4 x^4}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^5,x]

[Out]

-a^8/(4*x^4) - (4*a^7*b)/x^2 + 28*a^5*b^3*x^2 + (35*a^4*b^4*x^4)/2 + (28*a^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 +
(4*a*b^7*x^10)/5 + (b^8*x^12)/12 + 28*a^6*b^2*Log[x]

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Maple [A]  time = 0.005, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{4\,{x}^{4}}}-4\,{\frac{{a}^{7}b}{{x}^{2}}}+28\,{a}^{5}{b}^{3}{x}^{2}+{\frac{35\,{a}^{4}{b}^{4}{x}^{4}}{2}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{6}}{3}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{8}}{2}}+{\frac{4\,a{b}^{7}{x}^{10}}{5}}+{\frac{{b}^{8}{x}^{12}}{12}}+28\,{a}^{6}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^5,x)

[Out]

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

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Maxima [A]  time = 1.94606, size = 124, normalized size = 1.23 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{4}{5} \, a b^{7} x^{10} + \frac{7}{2} \, a^{2} b^{6} x^{8} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac{16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^5,x, algorithm="maxima")

[Out]

1/12*b^8*x^12 + 4/5*a*b^7*x^10 + 7/2*a^2*b^6*x^8 + 28/3*a^3*b^5*x^6 + 35/2*a^4*b^4*x^4 + 28*a^5*b^3*x^2 + 14*a
^6*b^2*log(x^2) - 1/4*(16*a^7*b*x^2 + a^8)/x^4

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Fricas [A]  time = 1.26282, size = 224, normalized size = 2.22 \begin{align*} \frac{5 \, b^{8} x^{16} + 48 \, a b^{7} x^{14} + 210 \, a^{2} b^{6} x^{12} + 560 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} + 1680 \, a^{6} b^{2} x^{4} \log \left (x\right ) - 240 \, a^{7} b x^{2} - 15 \, a^{8}}{60 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^5,x, algorithm="fricas")

[Out]

1/60*(5*b^8*x^16 + 48*a*b^7*x^14 + 210*a^2*b^6*x^12 + 560*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 1680*a^5*b^3*x^6 +
1680*a^6*b^2*x^4*log(x) - 240*a^7*b*x^2 - 15*a^8)/x^4

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Sympy [A]  time = 0.424023, size = 102, normalized size = 1.01 \begin{align*} 28 a^{6} b^{2} \log{\left (x \right )} + 28 a^{5} b^{3} x^{2} + \frac{35 a^{4} b^{4} x^{4}}{2} + \frac{28 a^{3} b^{5} x^{6}}{3} + \frac{7 a^{2} b^{6} x^{8}}{2} + \frac{4 a b^{7} x^{10}}{5} + \frac{b^{8} x^{12}}{12} - \frac{a^{8} + 16 a^{7} b x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**5,x)

[Out]

28*a**6*b**2*log(x) + 28*a**5*b**3*x**2 + 35*a**4*b**4*x**4/2 + 28*a**3*b**5*x**6/3 + 7*a**2*b**6*x**8/2 + 4*a
*b**7*x**10/5 + b**8*x**12/12 - (a**8 + 16*a**7*b*x**2)/(4*x**4)

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Giac [A]  time = 1.89315, size = 139, normalized size = 1.38 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{4}{5} \, a b^{7} x^{10} + \frac{7}{2} \, a^{2} b^{6} x^{8} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac{84 \, a^{6} b^{2} x^{4} + 16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^5,x, algorithm="giac")

[Out]

1/12*b^8*x^12 + 4/5*a*b^7*x^10 + 7/2*a^2*b^6*x^8 + 28/3*a^3*b^5*x^6 + 35/2*a^4*b^4*x^4 + 28*a^5*b^3*x^2 + 14*a
^6*b^2*log(x^2) - 1/4*(84*a^6*b^2*x^4 + 16*a^7*b*x^2 + a^8)/x^4