3.225 \(\int \frac{x^5}{(a x^2+b x^3)^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

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Rubi [A]  time = 0.0180811, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 43} \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a*x^2 + b*x^3)^2,x]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0063073, size = 20, normalized size = 0.87 \[ \frac{\frac{a}{a+b x}+\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a*x^2 + b*x^3)^2,x]

[Out]

(a/(a + b*x) + Log[a + b*x])/b^2

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Maple [A]  time = 0.005, size = 24, normalized size = 1. \begin{align*}{\frac{a}{{b}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^3+a*x^2)^2,x)

[Out]

a/b^2/(b*x+a)+ln(b*x+a)/b^2

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Maxima [A]  time = 0.983391, size = 35, normalized size = 1.52 \begin{align*} \frac{a}{b^{3} x + a b^{2}} + \frac{\log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2)^2,x, algorithm="maxima")

[Out]

a/(b^3*x + a*b^2) + log(b*x + a)/b^2

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Fricas [A]  time = 0.765177, size = 62, normalized size = 2.7 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

((b*x + a)*log(b*x + a) + a)/(b^3*x + a*b^2)

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Sympy [A]  time = 0.527788, size = 20, normalized size = 0.87 \begin{align*} \frac{a}{a b^{2} + b^{3} x} + \frac{\log{\left (a + b x \right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**3+a*x**2)**2,x)

[Out]

a/(a*b**2 + b**3*x) + log(a + b*x)/b**2

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Giac [A]  time = 1.19752, size = 32, normalized size = 1.39 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{2}} + \frac{a}{{\left (b x + a\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

log(abs(b*x + a))/b^2 + a/((b*x + a)*b^2)