3.93 \(\int \frac{(a+b x^2)^8}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0599015, 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} \[ \frac{14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac{35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac{a^8}{2 x^2}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0044414, size = 99, normalized size = 1. \[ \frac{14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac{35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac{a^8}{2 x^2}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{2\,{x}^{2}}}+14\,{a}^{6}{b}^{2}{x}^{2}+14\,{a}^{5}{b}^{3}{x}^{4}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+7\,{a}^{3}{b}^{5}{x}^{8}+{\frac{14\,{a}^{2}{b}^{6}{x}^{10}}{5}}+{\frac{2\,a{b}^{7}{x}^{12}}{3}}+{\frac{{b}^{8}{x}^{14}}{14}}+8\,{a}^{7}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^3,x)

[Out]

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

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Maxima [A]  time = 1.75426, size = 123, normalized size = 1.24 \begin{align*} \frac{1}{14} \, b^{8} x^{14} + \frac{2}{3} \, a b^{7} x^{12} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac{35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac{a^{8}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50534, size = 232, normalized size = 2.34 \begin{align*} \frac{15 \, b^{8} x^{16} + 140 \, a b^{7} x^{14} + 588 \, a^{2} b^{6} x^{12} + 1470 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 2940 \, a^{5} b^{3} x^{6} + 2940 \, a^{6} b^{2} x^{4} + 1680 \, a^{7} b x^{2} \log \left (x\right ) - 105 \, a^{8}}{210 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.371833, size = 100, normalized size = 1.01 \begin{align*} - \frac{a^{8}}{2 x^{2}} + 8 a^{7} b \log{\left (x \right )} + 14 a^{6} b^{2} x^{2} + 14 a^{5} b^{3} x^{4} + \frac{35 a^{4} b^{4} x^{6}}{3} + 7 a^{3} b^{5} x^{8} + \frac{14 a^{2} b^{6} x^{10}}{5} + \frac{2 a b^{7} x^{12}}{3} + \frac{b^{8} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**3,x)

[Out]

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

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Giac [A]  time = 2.84298, size = 136, normalized size = 1.37 \begin{align*} \frac{1}{14} \, b^{8} x^{14} + \frac{2}{3} \, a b^{7} x^{12} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac{35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac{8 \, a^{7} b x^{2} + a^{8}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^3,x, algorithm="giac")

[Out]

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