Optimal. Leaf size=28 \[ 2 \tan ^{-1}\left (\frac {x \sqrt {a x^3-b}}{b-a x^3}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b+a x^3}}+\frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-b+a x^3}} \, dx+\int \frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}+\int \left (-\frac {3 b}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}}-\frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}-(3 b) \int \frac {1}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}} \, dx-\int \frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.25, size = 2752, normalized size = 98.29 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.55, size = 28, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x \sqrt {-b+a x^3}}{b-a x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 40, normalized size = 1.43 \begin {gather*} \arctan \left (\frac {{\left (a x^{3} - x^{2} - b\right )} \sqrt {a x^{3} - b}}{2 \, {\left (a x^{4} - b x\right )}}\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*} \int \frac {a x^{3} + 2 \, b}{{\left (a x^{3} + x^{2} - b\right )} \sqrt {a x^{3} - b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.22, size = 786, normalized size = 28.07
method | result | size |
default | \(\frac {2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}}{-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}}}\, \sqrt {\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{3 a \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+\textit {\_Z}^{2}-b \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 b \right ) \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i a \left (2 x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}-\underline {\hspace {1.25 ex}}\alpha \left (a^{2} b \right )^{\frac {2}{3}} a -\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +2 a^{2} b -\left (a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b +i \sqrt {3}\, a b +3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{2 a b}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a +2\right ) \sqrt {a \,x^{3}-b}}\right )}{a^{2} b}\) | \(786\) |
elliptic | \(\frac {2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}}{-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}}}\, \sqrt {\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{3 a \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+\textit {\_Z}^{2}-b \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 b \right ) \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i a \left (2 x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a -\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}-\underline {\hspace {1.25 ex}}\alpha \left (a^{2} b \right )^{\frac {2}{3}} a -\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a +2 a^{2} b -\left (a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +i \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 i \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b +i \sqrt {3}\, a b +3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{2 a b}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a +2\right ) \sqrt {a \,x^{3}-b}}\right )}{a^{2} b}\) | \(786\) |
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*} \int \frac {a x^{3} + 2 \, b}{{\left (a x^{3} + x^{2} - b\right )} \sqrt {a x^{3} - b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.06, size = 45, normalized size = 1.61 \begin {gather*} \ln \left (\frac {b-a\,x^3+x^2-x\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{a\,x^3+x^2-b}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________