Optimal. Leaf size=53 \[ -\frac {a \log \left (a+b x^3\right )}{3 b (b c-a d)}+\frac {c \log \left (c+d x^3\right )}{3 d (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78}
\begin {gather*} \frac {c \log \left (c+d x^3\right )}{3 d (b c-a d)}-\frac {a \log \left (a+b x^3\right )}{3 b (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{(a+b x) (c+d x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a}{(b c-a d) (a+b x)}+\frac {c}{(b c-a d) (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \log \left (a+b x^3\right )}{3 b (b c-a d)}+\frac {c \log \left (c+d x^3\right )}{3 d (b c-a d)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 43, normalized size = 0.81 \begin {gather*} -\frac {a d \log \left (a+b x^3\right )-b c \log \left (c+d x^3\right )}{3 b^2 c d-3 a b d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.34, size = 50, normalized size = 0.94
method | result | size |
default | \(\frac {a \ln \left (b \,x^{3}+a \right )}{3 \left (a d -b c \right ) b}-\frac {c \ln \left (d \,x^{3}+c \right )}{3 \left (a d -b c \right ) d}\) | \(50\) |
norman | \(\frac {a \ln \left (b \,x^{3}+a \right )}{3 \left (a d -b c \right ) b}-\frac {c \ln \left (d \,x^{3}+c \right )}{3 \left (a d -b c \right ) d}\) | \(50\) |
risch | \(-\frac {c \ln \left (-d \,x^{3}-c \right )}{3 d \left (a d -b c \right )}+\frac {a \ln \left (b \,x^{3}+a \right )}{3 \left (a d -b c \right ) b}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 49, normalized size = 0.92 \begin {gather*} -\frac {a \log \left (b x^{3} + a\right )}{3 \, {\left (b^{2} c - a b d\right )}} + \frac {c \log \left (d x^{3} + c\right )}{3 \, {\left (b c d - a d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.37, size = 42, normalized size = 0.79 \begin {gather*} -\frac {a d \log \left (b x^{3} + a\right ) - b c \log \left (d x^{3} + c\right )}{3 \, {\left (b^{2} c d - a b d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (39) = 78\).
time = 5.88, size = 144, normalized size = 2.72 \begin {gather*} \frac {a \log {\left (x^{3} + \frac {\frac {a^{3} d^{2}}{b \left (a d - b c\right )} - \frac {2 a^{2} c d}{a d - b c} + \frac {a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{3 b \left (a d - b c\right )} - \frac {c \log {\left (x^{3} + \frac {- \frac {a^{2} c d}{a d - b c} + \frac {2 a b c^{2}}{a d - b c} + 2 a c - \frac {b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{3 d \left (a d - b c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.63, size = 51, normalized size = 0.96 \begin {gather*} -\frac {a \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, {\left (b^{2} c - a b d\right )}} + \frac {c \log \left ({\left | d x^{3} + c \right |}\right )}{3 \, {\left (b c d - a d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.31, size = 51, normalized size = 0.96 \begin {gather*} -\frac {a\,\ln \left (b\,x^3+a\right )}{3\,b^2\,c-3\,a\,b\,d}-\frac {c\,\ln \left (d\,x^3+c\right )}{3\,a\,d^2-3\,b\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________