Optimal. Leaf size=277 \[ \frac {4 \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {b x^3-a}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2139, 219, 2138, 203} \[ \frac {4 \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {b x^3-a}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 219
Rule 2138
Rule 2139
Rubi steps
\begin {align*} \int \frac {x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx &=\frac {\int \frac {1}{\sqrt {-a+b x^3}} \, dx}{3 \sqrt [3]{b}}-\frac {\int \frac {2 \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx}{3 \sqrt [3]{b}}\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \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 )}{3 \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 {\left (4 \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{9+a x^2} \, dx,x,\frac {\left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt {-a+b x^3}}\right )}{3 b^{2/3}}\\ &=\frac {4 \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a+b x^3}}\right )}{9 \sqrt [6]{a} b^{2/3}}-\frac {2 \sqrt {2-\sqrt {3}} \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 )}{3 \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 = 0.23, size = 372, normalized size = 1.34 \[ \frac {2 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )+\frac {2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {b^{2/3} x^2}{a^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {b x^3-a}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b^{\frac {1}{3}} x +2 a^{\frac {1}{3}}\right ) \sqrt {b \,x^{3}-a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b x^{3} - a} {\left (b^{\frac {1}{3}} x + 2 \, a^{\frac {1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{\left (b^{1/3}\,x+2\,a^{1/3}\right )\,\sqrt {b\,x^3-a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (2 \sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt {- a + b x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________