Optimal. Leaf size=176 \[ \frac {\left (c \sqrt {b-a}-3 a b \sqrt {b-a}\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-a}}\right )}{3 a \sqrt {b}}+\frac {\left (-c \sqrt {a+b}-3 a b \sqrt {a+b}\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}+\frac {2}{3} \sqrt {a^2 x^3+b^2}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right ) \]
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Rubi [A] time = 0.78, antiderivative size = 223, normalized size of antiderivative = 1.27, number of steps used = 14, number of rules used = 7, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6725, 266, 50, 63, 208, 444, 205} \begin {gather*} \frac {(3 a b-c) \sqrt {a^2 x^3+b^2}}{3 a b}+\frac {(3 a b+c) \sqrt {a^2 x^3+b^2}}{3 a b}-\frac {\sqrt {a-b} (3 a b-c) \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 a \sqrt {b}}-\frac {\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}-\frac {4}{3} \sqrt {a^2 x^3+b^2}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 208
Rule 266
Rule 444
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {b^2+a^2 x^3}}{x}-\frac {(3 a b+c) x^2 \sqrt {b^2+a^2 x^3}}{2 b \left (b-a x^3\right )}+\frac {(3 a b-c) x^2 \sqrt {b^2+a^2 x^3}}{2 b \left (b+a x^3\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx\right )+\frac {(3 a b-c) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b+a x^3} \, dx}{2 b}-\frac {(3 a b+c) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b-a x^3} \, dx}{2 b}\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )\right )+\frac {(3 a b-c) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b+a x} \, dx,x,x^3\right )}{6 b}-\frac {(3 a b+c) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b-a x} \, dx,x,x^3\right )}{6 b}\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {1}{3} \left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{6} ((a-b) (3 a b-c)) \operatorname {Subst}\left (\int \frac {1}{(b+a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{6} ((a+b) (3 a b+c)) \operatorname {Subst}\left (\int \frac {1}{(b-a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {\left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}-\frac {((a-b) (3 a b-c)) \operatorname {Subst}\left (\int \frac {1}{b-\frac {b^2}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}-\frac {((a+b) (3 a b+c)) \operatorname {Subst}\left (\int \frac {1}{b+\frac {b^2}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 a^2}\\ &=-\frac {4}{3} \sqrt {b^2+a^2 x^3}+\frac {(3 a b-c) \sqrt {b^2+a^2 x^3}}{3 a b}+\frac {(3 a b+c) \sqrt {b^2+a^2 x^3}}{3 a b}-\frac {\sqrt {a-b} (3 a b-c) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 a \sqrt {b}}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )-\frac {\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 194, normalized size = 1.10 \begin {gather*} \frac {c \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )-\sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )+2 a \sqrt {b} \sqrt {a^2 x^3+b^2}-3 a b \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )+4 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 a \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 176, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {b^2+a^2 x^3}+\frac {\left (-3 a \sqrt {a-b} b+\sqrt {a-b} c\right ) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 a \sqrt {b}}+\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\left (-3 a b \sqrt {a+b}-\sqrt {a+b} c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 735, normalized size = 4.18 \begin {gather*} \left [\frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {2 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} a}{3 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 183, normalized size = 1.04 \begin {gather*} \frac {2}{3} \, b \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) - \frac {2}{3} \, b \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) + \frac {2}{3} \, \sqrt {a^{2} x^{3} + b^{2}} - \frac {{\left (3 \, a^{2} b - 3 \, a b^{2} - a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} a} + \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.79, size = 943, normalized size = 5.36
method | result | size |
default | \(\frac {\left (3 a b +c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a +b \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b}+\frac {\left (3 a b -c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a -b \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b}-\frac {4 \sqrt {a^{2} x^{3}+b^{2}}}{3}+\frac {4 b^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\) | \(943\) |
elliptic | \(\text {Expression too large to display}\) | \(3175\) |
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 {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} - b^{2}\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.52, size = 204, normalized size = 1.16 \begin {gather*} \frac {2\,\sqrt {a^2\,x^3+b^2}}{3}+\frac {2\,b\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )}{3}+\frac {\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,a\,\sqrt {b}}+\frac {\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a-b}\,2{}\mathrm {i}}{a\,x^3+b}\right )\,\sqrt {a-b}\,\left (c-3\,a\,b\right )\,1{}\mathrm {i}}{6\,a\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 114.32, size = 162, normalized size = 0.92 \begin {gather*} - \frac {2 b \log {\left (b - \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {2 b \log {\left (b + \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {2 \sqrt {a^{2} x^{3} + b^{2}}}{3} - \frac {\left (a - b\right ) \left (3 a b - c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}} \right )}}{3 a \sqrt {a b - b^{2}}} + \frac {\left (a + b\right ) \left (3 a b + c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {- a b - b^{2}}} \right )}}{3 a \sqrt {- a b - b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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