Optimal. Leaf size=33 \[ \frac {x \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac {5}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251}
\begin {gather*} \frac {x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 252
Rubi steps
\begin {align*} \int \sqrt [3]{a+b x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \sqrt [3]{1+\frac {b x^3}{a}} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.17, size = 196, normalized size = 5.94 \begin {gather*} \frac {3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} F_1\left (\frac {4}{3};-\frac {1}{3},-\frac {1}{3};\frac {7}{3};-\frac {i \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {3} \sqrt [3]{a}},\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}}{3 i+\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \right )^{\frac {1}{3}}\, dx\]
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 [F]
time = 0.37, size = 11, normalized size = 0.33 \begin {gather*} {\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac {1}{3}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.39, size = 37, normalized size = 1.12 \begin {gather*} \frac {\sqrt [3]{a} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.01, size = 37, normalized size = 1.12 \begin {gather*} \frac {x\,{\left (b\,x^3+a\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (\frac {b\,x^3}{a}+1\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________