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

Optimal. Leaf size=100 \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]

[Out]

4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)
/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^8*Log[x]

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Rubi [A]  time = 0.058515, antiderivative size = 100, 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} \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x,x]

[Out]

4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)
/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^8*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} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (8 a^7 b+\frac{a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^2\right )\\ &=4 a^7 b x^2+7 a^6 b^2 x^4+\frac{28}{3} a^5 b^3 x^6+\frac{35}{4} a^4 b^4 x^8+\frac{28}{5} a^3 b^5 x^{10}+\frac{7}{3} a^2 b^6 x^{12}+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16}+a^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0041288, size = 100, normalized size = 1. \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x,x]

[Out]

4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)
/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^8*Log[x]

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Maple [A]  time = 0.003, size = 89, normalized size = 0.9 \begin{align*} 4\,{a}^{7}b{x}^{2}+7\,{a}^{6}{b}^{2}{x}^{4}+{\frac{28\,{a}^{5}{b}^{3}{x}^{6}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{8}}{4}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{10}}{5}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{12}}{3}}+{\frac{4\,a{b}^{7}{x}^{14}}{7}}+{\frac{{b}^{8}{x}^{16}}{16}}+{a}^{8}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.70948, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac{1}{2} \, a^{8} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48633, size = 205, normalized size = 2.05 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.324676, size = 102, normalized size = 1.02 \begin{align*} a^{8} \log{\left (x \right )} + 4 a^{7} b x^{2} + 7 a^{6} b^{2} x^{4} + \frac{28 a^{5} b^{3} x^{6}}{3} + \frac{35 a^{4} b^{4} x^{8}}{4} + \frac{28 a^{3} b^{5} x^{10}}{5} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{4 a b^{7} x^{14}}{7} + \frac{b^{8} x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*log(x) + 4*a**7*b*x**2 + 7*a**6*b**2*x**4 + 28*a**5*b**3*x**6/3 + 35*a**4*b**4*x**8/4 + 28*a**3*b**5*x**1
0/5 + 7*a**2*b**6*x**12/3 + 4*a*b**7*x**14/7 + b**8*x**16/16

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Giac [A]  time = 2.4002, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac{1}{2} \, a^{8} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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