3.4.54 \(\int \frac {x^2}{a-b x^3} \, dx\) [354]

Optimal. Leaf size=16 \[ -\frac {\log \left (a-b x^3\right )}{3 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {266} \begin {gather*} -\frac {\log \left (a-b x^3\right )}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^3),x]

[Out]

-1/3*Log[a - b*x^3]/b

Rule 266

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a-b x^3\right )}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^3),x]

[Out]

-1/3*Log[a - b*x^3]/b

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Maple [A]
time = 0.12, size = 15, normalized size = 0.94

method	result	size
derivativedivides	\(-\frac {\ln \left (-b \,x^{3}+a \right )}{3 b}\)	\(15\)
default	\(-\frac {\ln \left (-b \,x^{3}+a \right )}{3 b}\)	\(15\)
norman	\(-\frac {\ln \left (-b \,x^{3}+a \right )}{3 b}\)	\(15\)
risch	\(-\frac {\ln \left (-b \,x^{3}+a \right )}{3 b}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 15, normalized size = 0.94 \begin {gather*} -\frac {\log \left (b x^{3} - a\right )}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(b*x^3 - a)/b

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Fricas [A]
time = 0.35, size = 15, normalized size = 0.94 \begin {gather*} -\frac {\log \left (b x^{3} - a\right )}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(b*x^3 - a)/b

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Sympy [A]
time = 0.05, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\log {\left (- a + b x^{3} \right )}}{3 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-a + b*x**3)/(3*b)

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Giac [A]
time = 2.05, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | b x^{3} - a \right |}\right )}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(abs(b*x^3 - a))/b

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Mupad [B]
time = 0.04, size = 15, normalized size = 0.94 \begin {gather*} -\frac {\ln \left (b\,x^3-a\right )}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a - b*x^3),x)

[Out]

-log(b*x^3 - a)/(3*b)

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