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

Optimal. Leaf size=42 \[ -\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {b}{a^2 (a+b x)}-\frac {1}{a^2 x} \]

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, 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, 44} \begin {gather*} -\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {1}{a^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a*x^2 + b*x^3)^2,x]

[Out]

-(1/(a^2*x)) - b/(a^2*(a + b*x)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x])/a^3

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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

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Mathematica [A]  time = 0.05, size = 35, normalized size = 0.83 \begin {gather*} -\frac {a \left (\frac {b}{a+b x}+\frac {1}{x}\right )-2 b \log (a+b x)+2 b \log (x)}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a*x^2 + b*x^3)^2,x]

[Out]

-((a*(x^(-1) + b/(a + b*x)) + 2*b*Log[x] - 2*b*Log[a + b*x])/a^3)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 63, normalized size = 1.50 \begin {gather*} -\frac {2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \relax (x)}{a^{3} b x^{2} + a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log(b*x + a) + 2*(b^2*x^2 + a*b*x)*log(x))/(a^3*b*x^2 + a^4*x)

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giac [A]  time = 0.15, size = 45, normalized size = 1.07 \begin {gather*} \frac {2 \, b \log \left ({\left | b x + a \right |}\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, b x + a}{{\left (b x^{2} + a x\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 43, normalized size = 1.02 \begin {gather*} -\frac {b}{\left (b x +a \right ) a^{2}}-\frac {2 b \ln \relax (x )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}-\frac {1}{a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2/x-b/a^2/(b*x+a)-2/a^3*b*ln(x)+2*b*ln(b*x+a)/a^3

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maxima [A]  time = 1.31, size = 45, normalized size = 1.07 \begin {gather*} -\frac {2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \log \relax (x)}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.34, size = 41, normalized size = 0.98 \begin {gather*} \frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3}-\frac {\frac {1}{a}+\frac {2\,b\,x}{a^2}}{b\,x^2+a\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x^2 + b*x^3)^2,x)

[Out]

(4*b*atanh((2*b*x)/a + 1))/a^3 - (1/a + (2*b*x)/a^2)/(a*x + b*x^2)

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sympy [A]  time = 0.27, size = 37, normalized size = 0.88 \begin {gather*} \frac {- a - 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac {2 b \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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