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 = 0.0327931, 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 \[ -\frac{d \log (x)}{c^2}+\frac{d \log (c+d x)}{c^2}-\frac{1}{c x} \]

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

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Rubi in Sympy [A]  time = 14.1775, size = 24, normalized size = 0.86 \[ - \frac{1}{c x} - \frac{d \log{\left (x \right )}}{c^{2}} + \frac{d \log{\left (c + d x \right )}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x**3+c*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.00777943, 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 = 0.014, size = 29, normalized size = 1. \[ -{\frac{1}{cx}}-{\frac{d\ln \left ( x \right ) }{{c}^{2}}}+{\frac{d\ln \left ( dx+c \right ) }{{c}^{2}}} \]

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 = 0.765276, size = 38, normalized size = 1.36 \[ \frac{d \log \left (d x + c\right )}{c^{2}} - \frac{d \log \left (x\right )}{c^{2}} - \frac{1}{c x} \]

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 = 0.317002, size = 35, normalized size = 1.25 \[ \frac{d x \log \left (d x + c\right ) - d x \log \left (x\right ) - c}{c^{2} x} \]

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 = 1.27418, size = 19, normalized size = 0.68 \[ - \frac{1}{c x} + \frac{d \left (- \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )}\right )}{c^{2}} \]

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/XCAS [A]  time = 0.259568, size = 41, normalized size = 1.46 \[ \frac{d{\rm ln}\left ({\left | d x + c \right |}\right )}{c^{2}} - \frac{d{\rm ln}\left ({\left | x \right |}\right )}{c^{2}} - \frac{1}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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