Optimal. Leaf size=19 \[ \frac {1}{3} \log \left (a-b x^3-c x^6\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1482, 642}
\begin {gather*} \frac {1}{3} \log \left (a-b x^3-c x^6\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 642
Rule 1482
Rubi steps
\begin {align*} \int \frac {x^2 \left (b+2 c x^3\right )}{-a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {b+2 c x}{-a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \log \left (a-b x^3-c x^6\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (-a+b x^3+c x^6\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 18, normalized size = 0.95
method | result | size |
default | \(\frac {\ln \left (-c \,x^{6}-b \,x^{3}+a \right )}{3}\) | \(18\) |
norman | \(\frac {\ln \left (-c \,x^{6}-b \,x^{3}+a \right )}{3}\) | \(18\) |
risch | \(\frac {\ln \left (-c \,x^{6}-b \,x^{3}+a \right )}{3}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, \log \left (c x^{6} + b x^{3} - a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, \log \left (c x^{6} + b x^{3} - a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.21, size = 14, normalized size = 0.74 \begin {gather*} \frac {\log {\left (- a + b x^{3} + c x^{6} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.08, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{3} \, \log \left ({\left | c x^{6} + b x^{3} - a \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} \frac {\ln \left (c\,x^6+b\,x^3-a\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________