Optimal. Leaf size=21 \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1586, 31} \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 1586
Rubi steps
\begin {align*} \int \frac {a^{2/3} C-\sqrt [3]{a} \sqrt [3]{b} C x+b^{2/3} C x^2}{a+b x^3} \, dx &=\int \frac {1}{\frac {\sqrt [3]{a}}{C}+\frac {\sqrt [3]{b} x}{C}} \, dx\\ &=\frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 17, normalized size = 0.81 \[ \frac {C \log \left (b x + a^{\frac {1}{3}} b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 16, normalized size = 0.76 \[ \frac {C \log \left ({\left | b^{\frac {1}{3}} x + a^{\frac {1}{3}} \right |}\right )}{b^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 218, normalized size = 10.38 \[ \frac {\sqrt {3}\, C \,a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {C \,a^{\frac {2}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {C \,a^{\frac {2}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\sqrt {3}\, C \,a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}+\frac {C \,a^{\frac {1}{3}} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}-\frac {C \,a^{\frac {1}{3}} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{\frac {2}{3}}}+\frac {C \ln \left (b \,x^{3}+a \right )}{3 b^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.99, size = 210, normalized size = 10.00 \[ -\frac {\sqrt {3} {\left (2 \, C a b^{\frac {2}{3}} + {\left (3 \, C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, C a}{b^{\frac {1}{3}}}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (2 \, C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} - C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - C a^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C b^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {2}{3}} + C a^{\frac {1}{3}} b^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + C a^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.90, size = 15, normalized size = 0.71 \[ \frac {C\,\ln \left (x+\frac {a^{1/3}}{b^{1/3}}\right )}{b^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.26, size = 20, normalized size = 0.95 \[ \frac {C \log {\left (\sqrt [3]{a} b^{\frac {2}{3}} + b x \right )}}{\sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________