3.42 \(\int \frac{B^2+B C x+C^2 x^2}{-B^3+C^3 x^3} \, dx\)

Optimal. Leaf size=11 \[ \frac{\log (B-C x)}{C} \]

[Out]

Log[B - C*x]/C

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Rubi [A]  time = 0.0105686, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {1586, 31} \[ \frac{\log (B-C x)}{C} \]

Antiderivative was successfully verified.

[In]

Int[(B^2 + B*C*x + C^2*x^2)/(-B^3 + C^3*x^3),x]

[Out]

Log[B - C*x]/C

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{B^2+B C x+C^2 x^2}{-B^3+C^3 x^3} \, dx &=\int \frac{1}{-B+C x} \, dx\\ &=\frac{\log (B-C x)}{C}\\ \end{align*}

Mathematica [A]  time = 0.0015161, size = 12, normalized size = 1.09 \[ \frac{\log (C x-B)}{C} \]

Antiderivative was successfully verified.

[In]

Integrate[(B^2 + B*C*x + C^2*x^2)/(-B^3 + C^3*x^3),x]

[Out]

Log[-B + C*x]/C

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Maple [A]  time = 0.039, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( -Cx+B \right ) }{C}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C^2*x^2+B*C*x+B^2)/(C^3*x^3-B^3),x)

[Out]

ln(-C*x+B)/C

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Maxima [A]  time = 0.99526, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left (C x - B\right )}{C} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C^2*x^2+B*C*x+B^2)/(C^3*x^3-B^3),x, algorithm="maxima")

[Out]

log(C*x - B)/C

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Fricas [A]  time = 0.918183, size = 22, normalized size = 2. \begin{align*} \frac{\log \left (C x - B\right )}{C} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C^2*x^2+B*C*x+B^2)/(C^3*x^3-B^3),x, algorithm="fricas")

[Out]

log(C*x - B)/C

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Sympy [A]  time = 0.068666, size = 7, normalized size = 0.64 \begin{align*} \frac{\log{\left (- B + C x \right )}}{C} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C**2*x**2+B*C*x+B**2)/(C**3*x**3-B**3),x)

[Out]

log(-B + C*x)/C

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Giac [A]  time = 1.50249, size = 18, normalized size = 1.64 \begin{align*} \frac{\log \left ({\left | C x - B \right |}\right )}{C} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C^2*x^2+B*C*x+B^2)/(C^3*x^3-B^3),x, algorithm="giac")

[Out]

log(abs(C*x - B))/C