Integrand size = 32, antiderivative size = 533 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {a-b x^3}}{b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1892, 224, 1891} \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {2 \sqrt {2+\sqrt {3}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{\frac {b}{a}}\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{\frac {b}{a}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \sqrt [3]{\frac {b}{a}} \sqrt {a-b x^3}}{b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]
[In]
[Out]
Rule 224
Rule 1891
Rule 1892
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.17 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {x \sqrt {1-\frac {b x^3}{a}} \left (-2 \left (1+\sqrt {3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {b x^3}{a}\right )+\sqrt [3]{\frac {b}{a}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {b x^3}{a}\right )\right )}{2 \sqrt {a-b x^3}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (399 ) = 798\).
Time = 1.78 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.78
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.11 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=-\frac {2 \, {\left (\sqrt {-b} {\left (\sqrt {3} + 1\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) + \sqrt {-b} \left (\frac {b}{a}\right )^{\frac {1}{3}} {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \]
[In]
[Out]
Time = 1.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.24 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=- \frac {x^{2} \sqrt [3]{\frac {b}{a}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} + \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {\sqrt {3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \]
[In]
[Out]
\[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=\int { -\frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} - 1}{\sqrt {-b x^{3} + a}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}} \, dx=\int \frac {\sqrt {3}-x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {a-b\,x^3}} \,d x \]
[In]
[Out]