3.24 \(\int \frac{1}{c x^2+d x^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]

[Out]

-(1/(c*x)) - (d*Log[x])/c^2 + (d*Log[c + d*x])/c^2

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Rubi [A]  time = 0.0151371, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2 + d*x^3)^(-1),x]

[Out]

-(1/(c*x)) - (d*Log[x])/c^2 + (d*Log[c + d*x])/c^2

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.004111, size = 28, normalized size = 1. \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2 + d*x^3)^(-1),x]

[Out]

-(1/(c*x)) - (d*Log[x])/c^2 + (d*Log[c + d*x])/c^2

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Maple [A]  time = 0.007, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{cx}}-{\frac{d\ln \left ( x \right ) }{{c}^{2}}}+{\frac{d\ln \left ( dx+c \right ) }{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x^3+c*x^2),x)

[Out]

-1/c/x-d*ln(x)/c^2+d*ln(d*x+c)/c^2

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Maxima [A]  time = 1.14695, size = 38, normalized size = 1.36 \begin{align*} \frac{d \log \left (d x + c\right )}{c^{2}} - \frac{d \log \left (x\right )}{c^{2}} - \frac{1}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x^3+c*x^2),x, algorithm="maxima")

[Out]

d*log(d*x + c)/c^2 - d*log(x)/c^2 - 1/(c*x)

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Fricas [A]  time = 1.22864, size = 61, normalized size = 2.18 \begin{align*} \frac{d x \log \left (d x + c\right ) - d x \log \left (x\right ) - c}{c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x^3+c*x^2),x, algorithm="fricas")

[Out]

(d*x*log(d*x + c) - d*x*log(x) - c)/(c^2*x)

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Sympy [A]  time = 0.435983, size = 19, normalized size = 0.68 \begin{align*} - \frac{1}{c x} + \frac{d \left (- \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )}\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x**3+c*x**2),x)

[Out]

-1/(c*x) + d*(-log(x) + log(c/d + x))/c**2

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Giac [A]  time = 1.26542, size = 41, normalized size = 1.46 \begin{align*} \frac{d \log \left ({\left | d x + c \right |}\right )}{c^{2}} - \frac{d \log \left ({\left | x \right |}\right )}{c^{2}} - \frac{1}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x^3+c*x^2),x, algorithm="giac")

[Out]

d*log(abs(d*x + c))/c^2 - d*log(abs(x))/c^2 - 1/(c*x)