3.226 \(\int \frac{x^4}{(a x^2+b x^3)^2} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{b (a+b x)} \]

[Out]

-(1/(b*(a + b*x)))

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Rubi [A]  time = 0.0073729, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 32} \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a*x^2 + b*x^3)^2,x]

[Out]

-(1/(b*(a + b*x)))

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^4}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac{1}{(a+b x)^2} \, dx\\ &=-\frac{1}{b (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0026073, size = 12, normalized size = 1. \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a*x^2 + b*x^3)^2,x]

[Out]

-(1/(b*(a + b*x)))

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \begin{align*} -{\frac{1}{b \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^3+a*x^2)^2,x)

[Out]

-1/b/(b*x+a)

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Maxima [A]  time = 0.988879, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a*x^2)^2,x, algorithm="maxima")

[Out]

-1/(b^2*x + a*b)

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Fricas [A]  time = 0.789475, size = 24, normalized size = 2. \begin{align*} -\frac{1}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 0.920552, size = 10, normalized size = 0.83 \begin{align*} - \frac{1}{a b + b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**3+a*x**2)**2,x)

[Out]

-1/(a*b + b**2*x)

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Giac [A]  time = 1.1492, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

-1/((b*x + a)*b)