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

Optimal. Leaf size=64 \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/(2*x^2) + 5*a^2*b^3*x^2 + (5*a*b^4*x^4)/4 + (b^5*x^6)/6 + 10*a^3*b^2*Log[x]

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Rubi [A]  time = 0.0361221, antiderivative size = 64, 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} \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^5,x]

[Out]

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

Mathematica [A]  time = 0.006366, size = 64, normalized size = 1. \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/(2*x^2) + 5*a^2*b^3*x^2 + (5*a*b^4*x^4)/4 + (b^5*x^6)/6 + 10*a^3*b^2*Log[x]

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Maple [A]  time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}+5\,{a}^{2}{b}^{3}{x}^{2}+{\frac{5\,a{b}^{4}{x}^{4}}{4}}+{\frac{{b}^{5}{x}^{6}}{6}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^5/x^4-5/2*a^4*b/x^2+5*a^2*b^3*x^2+5/4*a*b^4*x^4+1/6*b^5*x^6+10*a^3*b^2*ln(x)

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Maxima [A]  time = 2.05834, size = 80, normalized size = 1.25 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac{10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^5*x^6 + 5/4*a*b^4*x^4 + 5*a^2*b^3*x^2 + 5*a^3*b^2*log(x^2) - 1/4*(10*a^4*b*x^2 + a^5)/x^4

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Fricas [A]  time = 1.37146, size = 139, normalized size = 2.17 \begin{align*} \frac{2 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} \log \left (x\right ) - 30 \, a^{4} b x^{2} - 3 \, a^{5}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^5*x^10 + 15*a*b^4*x^8 + 60*a^2*b^3*x^6 + 120*a^3*b^2*x^4*log(x) - 30*a^4*b*x^2 - 3*a^5)/x^4

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Sympy [A]  time = 0.486586, size = 61, normalized size = 0.95 \begin{align*} 10 a^{3} b^{2} \log{\left (x \right )} + 5 a^{2} b^{3} x^{2} + \frac{5 a b^{4} x^{4}}{4} + \frac{b^{5} x^{6}}{6} - \frac{a^{5} + 10 a^{4} b x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a**3*b**2*log(x) + 5*a**2*b**3*x**2 + 5*a*b**4*x**4/4 + b**5*x**6/6 - (a**5 + 10*a**4*b*x**2)/(4*x**4)

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Giac [A]  time = 2.1646, size = 95, normalized size = 1.48 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac{30 \, a^{3} b^{2} x^{4} + 10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^5*x^6 + 5/4*a*b^4*x^4 + 5*a^2*b^3*x^2 + 5*a^3*b^2*log(x^2) - 1/4*(30*a^3*b^2*x^4 + 10*a^4*b*x^2 + a^5)/x
^4