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

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a*x^2 + b*x^3)^2,x]

[Out]

-(1/(b*(a + b*x)))

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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b (a+b x)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\left (a x^2+b x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(a*x^2 + b*x^3)^2,x]

[Out]

IntegrateAlgebraic[x^4/(a*x^2 + b*x^3)^2, x]

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fricas [A]  time = 0.38, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}

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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giac [A]  time = 0.15, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b x + a\right )} b} \end {gather*}

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)

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maple [A]  time = 0.05, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{\left (b x +a \right ) b} \end {gather*}

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 = 1.30, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}

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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mupad [B]  time = 5.17, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a*x^2 + b*x^3)^2,x)

[Out]

-1/(b*(a + b*x))

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sympy [A]  time = 0.14, size = 10, normalized size = 0.83 \begin {gather*} - \frac {1}{a b + b^{2} x} \end {gather*}

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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