3.3.17 \(\int \frac {x^2}{a x^2+b x^3} \, dx\) [217]

Optimal. Leaf size=10 \[ \frac {\log (a+b x)}{b} \]

[Out]

ln(b*x+a)/b

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 31} \begin {gather*} \frac {\log (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

Rule 31

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

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a*x^2 + b*x^3),x]

[Out]

Log[a + b*x]/b

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Maple [A]
time = 0.34, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)
norman	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)
risch	\(\frac {\ln \left (b x +a \right )}{b}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(b*x+a)/b

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Maxima [A]
time = 0.28, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/b

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Fricas [A]
time = 1.58, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/b

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (a + b x \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x)/b

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Giac [A]
time = 2.61, size = 11, normalized size = 1.10 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x + a))/b

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Mupad [B]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x^2 + b*x^3),x)

[Out]

log(a + b*x)/b

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