3.2.81 \(\int \frac {x^3}{b x^2+c x^4} \, dx\) [181]

Optimal. Leaf size=15 \[ \frac {\log \left (b+c x^2\right )}{2 c} \]

[Out]

1/2*ln(c*x^2+b)/c

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Rubi [A]
time = 0.01, antiderivative size = 15, 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, 266} \begin {gather*} \frac {\log \left (b+c x^2\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^2 + c*x^4),x]

[Out]

Log[b + c*x^2]/(2*c)

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]

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^3}{b x^2+c x^4} \, dx &=\int \frac {x}{b+c x^2} \, dx\\ &=\frac {\log \left (b+c x^2\right )}{2 c}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b+c x^2\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b*x^2 + c*x^4),x]

[Out]

Log[b + c*x^2]/(2*c)

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Maple [A]
time = 0.09, size = 14, normalized size = 0.93

method	result	size
default	\(\frac {\ln \left (c \,x^{2}+b \right )}{2 c}\)	\(14\)
norman	\(\frac {\ln \left (c \,x^{2}+b \right )}{2 c}\)	\(14\)
risch	\(\frac {\ln \left (c \,x^{2}+b \right )}{2 c}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(c*x^2+b)/c

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (c x^{2} + b\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

1/2*log(c*x^2 + b)/c

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Fricas [A]
time = 0.33, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (c x^{2} + b\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

1/2*log(c*x^2 + b)/c

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (b + c x^{2} \right )}}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**4+b*x**2),x)

[Out]

log(b + c*x**2)/(2*c)

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Giac [A]
time = 5.18, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

1/2*log(abs(c*x^2 + b))/c

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )}{2\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2 + c*x^4),x)

[Out]

log(b + c*x^2)/(2*c)

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