Optimal. Leaf size=207 \[ \frac {\sqrt {a^2 x^3+b^2} \left (a^2 x^3+2 b^2+2 c x^3\right )}{6 b^2 x^6}-\frac {\sqrt {b-a} \left (3 a^2 b-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-a}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {\left (-a^4+12 a^2 b^2+2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \]
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Rubi [A] time = 0.91, antiderivative size = 359, normalized size of antiderivative = 1.73, number of steps used = 23, number of rules used = 9, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.173, Rules used = {6725, 266, 47, 51, 63, 208, 50, 444, 205} \begin {gather*} \frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {2 a^2 \sqrt {a^2 x^3+b^2}}{b^2}+\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{b}+\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}-\frac {a \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 b^{5/2}}-\frac {a \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 b^{5/2}}+\frac {a (3 a b-c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a (3 a b+c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}-\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 51
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^7 \left (-b^2+a^2 x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {b^2+a^2 x^3}}{x^7}-\frac {c \sqrt {b^2+a^2 x^3}}{b^2 x^4}-\frac {3 a^2 \sqrt {b^2+a^2 x^3}}{b^2 x}-\frac {a^2 (3 a b+c) x^2 \sqrt {b^2+a^2 x^3}}{2 b^3 \left (b-a x^3\right )}+\frac {a^2 (3 a b-c) x^2 \sqrt {b^2+a^2 x^3}}{2 b^3 \left (b+a x^3\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {b^2+a^2 x^3}}{x^7} \, dx\right )-\frac {\left (3 a^2\right ) \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx}{b^2}+\frac {\left (a^2 (3 a b-c)\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b+a x^3} \, dx}{2 b^3}-\frac {c \int \frac {\sqrt {b^2+a^2 x^3}}{x^4} \, dx}{b^2}-\frac {\left (a^2 (3 a b+c)\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^3}}{b-a x^3} \, dx}{2 b^3}\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^3} \, dx,x,x^3\right )\right )-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )}{b^2}+\frac {\left (a^2 (3 a b-c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b+a x} \, dx,x,x^3\right )}{6 b^3}-\frac {c \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^2} \, dx,x,x^3\right )}{3 b^2}-\frac {\left (a^2 (3 a b+c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{b-a x} \, dx,x,x^3\right )}{6 b^3}\\ &=-\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {1}{6} a^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {\left (a^2 (a-b) (3 a b-c)\right ) \operatorname {Subst}\left (\int \frac {1}{(b+a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}-\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}-\frac {\left (a^2 (a+b) (3 a b+c)\right ) \operatorname {Subst}\left (\int \frac {1}{(b-a x) \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}\\ &=-\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-2 \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 )+\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{12 b^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 b^2}-\frac {c \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 b^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 b^2}\\ &=-\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {a \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 b^{5/2}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{b}+\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a \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 b^{5/2}}+\frac {a^2 \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 )}{6 b^2}\\ &=-\frac {2 a^2 \sqrt {b^2+a^2 x^3}}{b^2}+\frac {a (3 a b-c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {a (3 a b+c) \sqrt {b^2+a^2 x^3}}{3 b^3}+\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}+\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}+\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {a \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 b^{5/2}}-\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{b}+\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a \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 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.09, size = 321, normalized size = 1.55 \begin {gather*} \frac {\frac {3 b^4 c \left (a^2 x^3 \sqrt {\frac {a^2 x^3}{b^2}+1} \tanh ^{-1}\left (\sqrt {\frac {a^2 x^3}{b^2}+1}\right )+a^2 x^3+b^2\right )}{x^3 \sqrt {a^2 x^3+b^2}}+18 a^2 b^4 \left (b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )-\sqrt {a^2 x^3+b^2}\right )-3 a b^3 (3 a b-c) \left (\sqrt {b} \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )-\sqrt {a^2 x^3+b^2}\right )-3 a b^3 (3 a b+c) \left (\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )-\sqrt {a^2 x^3+b^2}\right )+4 a^4 \left (a^2 x^3+b^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {a^2 x^3}{b^2}+1\right )}{9 b^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 207, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+a^2 x^3+2 c x^3\right )}{6 b^2 x^6}-\frac {\sqrt {a-b} \left (3 a^2 b-a c\right ) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (-a^4+12 a^2 b^2+2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 1051, normalized size = 5.08 \begin {gather*} \left [-\frac {2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \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 ) - 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - 2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 308, normalized size = 1.49 \begin {gather*} -\frac {{\left (3 \, a^{3} b - 3 \, a^{2} b^{2} - a^{2} c + a 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}} b^{2}} + \frac {{\left (3 \, a^{3} b + 3 \, a^{2} b^{2} + a^{2} c + a 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}} b^{2}} - \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {\sqrt {a^{2} x^{3} + b^{2}} a^{4} b^{2} + {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{4} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} a^{2} b^{2} c + 2 \, {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{2} c}{6 \, a^{4} b^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.71, size = 984, normalized size = 4.75
method | result | size |
risch | \(\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 x^{3} c +2 b^{2}\right )}{6 b^{2} x^{6}}+\frac {a^{2} \left (\frac {i \left (6 a^{2} b +6 a \,b^{2}+2 a c +2 b c \right ) \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 \left (a +b \right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2} b}-\frac {2 \left (a^{2}-12 b^{2}-2 c \right ) \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}-\frac {i \left (-6 a^{2} b +6 a \,b^{2}+2 a c -2 b c \right ) \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 \left (a -b \right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2} b}\right )}{4 b^{2}}\) | \(984\) |
default | \(-\frac {c \left (-\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{3}}-\frac {a^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}+\frac {a^{2} \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^{3}}+\frac {a^{2} \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^{3}}-\frac {3 a^{2} \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}-\frac {2 b^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}+\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{6}}+\frac {a^{2} \sqrt {a^{2} x^{3}+b^{2}}}{6 b^{2} x^{3}}-\frac {a^{4} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{6 b^{2} \sqrt {b^{2}}}\) | \(1088\) |
elliptic | \(\text {Expression too large to display}\) | \(3271\) |
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^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.50, size = 248, normalized size = 1.20 \begin {gather*} \frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a^2\,\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 )\,\left (-a^2+12\,b^2+2\,c\right )}{12\,b^3}+\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}+\frac {a\,\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\,b^{5/2}}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b-a}}{a\,x^3+b}\right )\,\sqrt {b-a}\,\left (c-3\,a\,b\right )}{6\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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