3.3.12 \(\int \frac {1}{a x^3+b x^4} \, dx\)

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

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 44} \begin {gather*} \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^4)^(-1),x]

[Out]

-1/(2*a*x^2) + b/(a^2*x) + (b^2*Log[x])/a^3 - (b^2*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 1593

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 42, normalized size = 1.00 \begin {gather*} \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^4)^(-1),x]

[Out]

-1/2*1/(a*x^2) + b/(a^2*x) + (b^2*Log[x])/a^3 - (b^2*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 {1}{a x^3+b x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*x^3 + b*x^4)^(-1),x]

[Out]

IntegrateAlgebraic[(a*x^3 + b*x^4)^(-1), x]

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fricas [A]  time = 0.39, size = 41, normalized size = 0.98 \begin {gather*} -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \relax (x) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 45, normalized size = 1.07 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 41, normalized size = 0.98 \begin {gather*} \frac {b^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}+\frac {b}{a^{2} x}-\frac {1}{2 a \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a*x^3),x)

[Out]

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

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maxima [A]  time = 1.36, size = 40, normalized size = 0.95 \begin {gather*} -\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \relax (x)}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.72, size = 38, normalized size = 0.90 \begin {gather*} -\frac {\frac {a^2}{2}-a\,b\,x}{a^3\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^3 + b*x^4),x)

[Out]

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

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sympy [A]  time = 0.22, size = 31, normalized size = 0.74 \begin {gather*} \frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \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(1/(b*x**4+a*x**3),x)

[Out]

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

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