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

Optimal. Leaf size=65 \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*Log[x]

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Rubi [A]  time = 0.0339468, antiderivative size = 65, 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} \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x,x]

[Out]

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

Mathematica [A]  time = 0.0038627, size = 65, normalized size = 1. \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x,x]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*Log[x]

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Maple [A]  time = 0.001, size = 56, normalized size = 0.9 \begin{align*}{\frac{5\,{a}^{4}b{x}^{2}}{2}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{x}^{10}}{10}}+{a}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*a^4*b*x^2+5/2*a^3*b^2*x^4+5/3*a^2*b^3*x^6+5/8*a*b^4*x^8+1/10*b^5*x^10+a^5*ln(x)

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Maxima [A]  time = 1.73627, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^2 + 1/2*a^5*log(x^2)

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Fricas [A]  time = 1.27972, size = 130, normalized size = 2. \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^2 + a^5*log(x)

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Sympy [A]  time = 0.304247, size = 65, normalized size = 1. \begin{align*} a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{2}}{2} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*log(x) + 5*a**4*b*x**2/2 + 5*a**3*b**2*x**4/2 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**8/8 + b**5*x**10/10

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Giac [A]  time = 2.15303, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*x^10 + 5/8*a*b^4*x^8 + 5/3*a^2*b^3*x^6 + 5/2*a^3*b^2*x^4 + 5/2*a^4*b*x^2 + 1/2*a^5*log(x^2)