Optimal. Leaf size=26 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a x^3+b}}\right )}{\sqrt {c}} \]
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Rubi [F] time = 1.18, 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-c x^2+a x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 b+a x^3}{\sqrt {b+a x^3} \left (b-c x^2+a x^3\right )} \, dx &=\int \left (\frac {1}{\sqrt {b+a x^3}}-\frac {3 b-c x^2}{\sqrt {b+a x^3} \left (b-c x^2+a x^3\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt {b+a x^3}} \, dx-\int \frac {3 b-c x^2}{\sqrt {b+a x^3} \left (b-c 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 {c x^2}{\left (-b+c x^2-a x^3\right ) \sqrt {b+a x^3}}+\frac {3 b}{\sqrt {b+a x^3} \left (b-c 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}{\sqrt {b+a x^3} \left (b-c x^2+a x^3\right )} \, dx-c \int \frac {x^2}{\left (-b+c x^2-a x^3\right ) \sqrt {b+a x^3}} \, dx\\ \end {align*}
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Mathematica [C] time = 6.28, size = 2764, normalized size = 106.31 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.57, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b+a x^3}}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 167, normalized size = 6.42 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} x^{6} + 6 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} + 6 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{4} + c x^{3} + b x\right )} \sqrt {a x^{3} + b} \sqrt {c}}{a^{2} x^{6} - 2 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} - 2 \, b c x^{2} + b^{2}}\right )}{2 \, \sqrt {c}}, \frac {\sqrt {-c} \arctan \left (\frac {{\left (a x^{3} + c x^{2} + b\right )} \sqrt {a x^{3} + b} \sqrt {-c}}{2 \, {\left (a c x^{4} + b c x\right )}}\right )}{c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} - 2 \, b}{{\left (a x^{3} - c x^{2} + b\right )} \sqrt {a x^{3} + b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 844, normalized size = 32.46
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}-c \,\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c -3 b \right ) \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}}}}\, \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 {-\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}}}}\, \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 c -i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, c +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\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 c -\left (-a^{2} b \right )^{\frac {2}{3}} c +2 a^{2} b \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 \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c +i \sqrt {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 ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -i \sqrt {3}\, a b c +3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 a b c}{2 a b c}, \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 c \right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b c}\) | \(844\) |
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}-c \,\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} c +3 b \right ) \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}}}}\, \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 {-\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}}}}\, \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 c -i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, c +\left (-a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\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 c -\left (-a^{2} b \right )^{\frac {2}{3}} c +2 a^{2} b \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 \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c +i \sqrt {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 ^{2} a -2 i \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, a b -i \sqrt {3}\, a b c +3 \left (-a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b +3 a b c}{2 a b c}, \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 c \right ) \sqrt {a \,x^{3}+b}}\right )}{a^{2} b c}\) | \(845\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} - 2 \, b}{{\left (a x^{3} - c x^{2} + b\right )} \sqrt {a x^{3} + b}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 40, normalized size = 1.54 \begin {gather*} \frac {\ln \left (\frac {\sqrt {c}\,x-\sqrt {a\,x^3+b}}{\sqrt {c}\,x+\sqrt {a\,x^3+b}}\right )}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} - 2 b}{\sqrt {a x^{3} + b} \left (a x^{3} + b - c x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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