Optimal. Leaf size=95 \[ \frac {3}{2} \sqrt [3]{(a+b x)^2}+\frac {a \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{\sqrt [3]{a^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 141, normalized size of antiderivative = 1.48, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1973, 52, 57,
632, 210, 31} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{(a+b x)^2} \arctan \left (\frac {2 \sqrt [3]{\frac {b x}{a}+1}+1}{\sqrt {3}}\right )}{\left (\frac {b x}{a}+1\right )^{2/3}}+\frac {3}{2} \sqrt [3]{(a+b x)^2}-\frac {\log (x) \sqrt [3]{(a+b x)^2}}{2 \left (\frac {b x}{a}+1\right )^{2/3}}+\frac {3 \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{\frac {b x}{a}+1}\right )}{2 \left (\frac {b x}{a}+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 52
Rule 57
Rule 210
Rule 632
Rule 1973
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\sqrt [3]{(a+b x)^2} \int \frac {\left (1+\frac {b x}{a}\right )^{2/3}}{x} \, dx}{\left (1+\frac {b x}{a}\right )^{2/3}}\\ &=\frac {3}{2} \sqrt [3]{(a+b x)^2}+\frac {\sqrt [3]{(a+b x)^2} \int \frac {1}{x \sqrt [3]{1+\frac {b x}{a}}} \, dx}{\left (1+\frac {b x}{a}\right )^{2/3}}\\ &=\frac {3}{2} \sqrt [3]{(a+b x)^2}-\frac {\sqrt [3]{(a+b x)^2} \log (x)}{2 \left (1+\frac {b x}{a}\right )^{2/3}}-\frac {\left (3 \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+\frac {b x}{a}}\right )}{2 \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+\frac {b x}{a}}\right )}{2 \left (1+\frac {b x}{a}\right )^{2/3}}\\ &=\frac {3}{2} \sqrt [3]{(a+b x)^2}-\frac {\sqrt [3]{(a+b x)^2} \log (x)}{2 \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {3 \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{1+\frac {b x}{a}}\right )}{2 \left (1+\frac {b x}{a}\right )^{2/3}}-\frac {\left (3 \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+\frac {b x}{a}}\right )}{\left (1+\frac {b x}{a}\right )^{2/3}}\\ &=\frac {3}{2} \sqrt [3]{(a+b x)^2}+\frac {\sqrt {3} \sqrt [3]{(a+b x)^2} \arctan \left (\frac {1+2 \sqrt [3]{1+\frac {b x}{a}}}{\sqrt {3}}\right )}{\left (1+\frac {b x}{a}\right )^{2/3}}-\frac {\sqrt [3]{(a+b x)^2} \log (x)}{2 \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {3 \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{1+\frac {b x}{a}}\right )}{2 \left (1+\frac {b x}{a}\right )^{2/3}}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 136, normalized size = 1.43 \begin {gather*} \frac {\sqrt [3]{(a+b x)^2} \left (3 (a+b x)^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{2 (a+b x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs.
\(2 (73) = 146\).
time = 0.60, size = 238, normalized size = 2.51 \begin {gather*} -\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (a b x + a^{2}\right )} + 2 \, \sqrt {3} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (a b x + a^{2}\right )}}\right ) - \frac {1}{2} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) + \frac {3}{2} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\left (a + b x\right )^{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{1/3}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________