Optimal. Leaf size=540 \[ \frac {2 \sqrt {2-\sqrt {3}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}-\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1880, 219, 1879} \[ \frac {2 \sqrt {2-\sqrt {3}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}-\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 219
Rule 1879
Rule 1880
Rubi steps
\begin {align*} \int \frac {1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\sqrt {-a-b x^3}} \, dx &=\frac {\sqrt [3]{\frac {b}{a}} \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {-a-b x^3}} \, dx}{\sqrt [3]{b}}+\left (1-\sqrt {3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {-a-b x^3}} \, dx\\ &=-\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {-a-b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}+\frac {2 \sqrt {2-\sqrt {3}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 92, normalized size = 0.17 \[ \frac {x \sqrt {\frac {b x^3}{a}+1} \left (x \sqrt [3]{\frac {b}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )-2 \left (\sqrt {3}-1\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a}\right )\right )}{2 \sqrt {-a-b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{3} - a} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {-b x^{3} - a} {\left (\sqrt {3} - 1\right )}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.09, size = 1013, normalized size = 1.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1}{\sqrt {-b x^{3} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1}{\sqrt {-b\,x^3-a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.70, size = 129, normalized size = 0.24 \[ - \frac {i x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} - \frac {i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {3} i x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________