3.3.29 \(\int \frac {x}{(a x^2+b x^3)^2} \, dx\) [229]

Optimal. Leaf size=58 \[ -\frac {1}{2 a^2 x^2}+\frac {2 b}{a^3 x}+\frac {b^2}{a^3 (a+b x)}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log (a+b x)}{a^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 46} \begin {gather*} \frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log (a+b x)}{a^4}+\frac {b^2}{a^3 (a+b x)}+\frac {2 b}{a^3 x}-\frac {1}{2 a^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a*x^2 + b*x^3)^2,x]

[Out]

-1/2*1/(a^2*x^2) + (2*b)/(a^3*x) + b^2/(a^3*(a + b*x)) + (3*b^2*Log[x])/a^4 - (3*b^2*Log[a + b*x])/a^4

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 1598

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

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Mathematica [A]
time = 0.04, size = 53, normalized size = 0.91 \begin {gather*} \frac {a \left (-\frac {a}{x^2}+\frac {4 b}{x}+\frac {2 b^2}{a+b x}\right )+6 b^2 \log (x)-6 b^2 \log (a+b x)}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a*x^2 + b*x^3)^2,x]

[Out]

(a*(-(a/x^2) + (4*b)/x + (2*b^2)/(a + b*x)) + 6*b^2*Log[x] - 6*b^2*Log[a + b*x])/(2*a^4)

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Maple [A]
time = 0.37, size = 57, normalized size = 0.98

method	result	size
default	\(-\frac {1}{2 a^{2} x^{2}}+\frac {2 b}{a^{3} x}+\frac {b^{2}}{a^{3} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}\)	\(57\)
risch	\(\frac {\frac {3 b^{2} x^{2}}{a^{3}}+\frac {3 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (-x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}\)	\(63\)
norman	\(\frac {-\frac {3 b^{3} x^{4}}{a^{4}}-\frac {x}{2 a}+\frac {3 b \,x^{2}}{2 a^{2}}}{x^{3} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 64, normalized size = 1.10 \begin {gather*} \frac {6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {3 \, b^{2} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, b^{2} \log \left (x\right )}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*b^2*x^2 + 3*a*b*x - a^2)/(a^3*b*x^3 + a^4*x^2) - 3*b^2*log(b*x + a)/a^4 + 3*b^2*log(x)/a^4

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Fricas [A]
time = 1.24, size = 86, normalized size = 1.48 \begin {gather*} \frac {6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} - 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.13, size = 54, normalized size = 0.93 \begin {gather*} \frac {- a^{2} + 3 a b x + 6 b^{2} x^{2}}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac {3 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**2 + 3*a*b*x + 6*b**2*x**2)/(2*a**4*x**2 + 2*a**3*b*x**3) + 3*b**2*(log(x) - log(a/b + x))/a**4

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Giac [A]
time = 1.77, size = 64, normalized size = 1.10 \begin {gather*} -\frac {3 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac {3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}}{2 \, {\left (b x + a\right )} a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b^2*log(abs(b*x + a))/a^4 + 3*b^2*log(abs(x))/a^4 + 1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3)/((b*x + a)*a^4*x^2)

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Mupad [B]
time = 5.31, size = 57, normalized size = 0.98 \begin {gather*} \frac {\frac {3\,b^2\,x^2}{a^3}-\frac {1}{2\,a}+\frac {3\,b\,x}{2\,a^2}}{b\,x^3+a\,x^2}-\frac {6\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*x^2 + b*x^3)^2,x)

[Out]

((3*b^2*x^2)/a^3 - 1/(2*a) + (3*b*x)/(2*a^2))/(a*x^2 + b*x^3) - (6*b^2*atanh((2*b*x)/a + 1))/a^4

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