3.1.64 \(\int \frac {x^2}{a+b x^3} \, dx\) [64]

Optimal. Leaf size=15 \[ \frac {\log \left (a+b x^3\right )}{3 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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]

Log[a + b*x^3]/(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 {gather*} \begin {aligned} \text {Integral} &=\frac {\log \left (a+b x^3\right )}{3 b}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 15, 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]

Log[a + b*x^3]/(3*b)

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Maple [A]
time = 0.01, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\)	\(14\)
default	\(\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\)	\(14\)
norman	\(\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\)	\(14\)
risch	\(\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\)	\(14\)
parallelrisch	\(\frac {\ln \left (b \,x^{3}+a \right )}{3 b}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 13, normalized size = 0.87 \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 = 0.56, size = 13, normalized size = 0.87 \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 = 10, normalized size = 0.67 \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 = 0.42, size = 14, normalized size = 0.93 \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 = 0.02, size = 13, normalized size = 0.87 \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(a + b*x^3)/(3*b)

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