Optimal. Leaf size=123 \[ -\frac {4 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}} \]
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Rubi [C] time = 3.39, antiderivative size = 747, normalized size of antiderivative = 6.07, number of steps used = 19, number of rules used = 11, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2056, 6715, 6725, 220, 2073, 1211, 1699, 208, 6728, 1217, 1707} \begin {gather*} -\frac {4 \sqrt {x} \sqrt {a^2 x^2+b^2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {2} \sqrt {x} \sqrt {a^2 x^2+b^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 \sqrt {a} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \left (3 \sqrt {-a^2}+\sqrt {3} a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {4 \sqrt {-a^2} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 220
Rule 1211
Rule 1217
Rule 1699
Rule 1707
Rule 2056
Rule 2073
Rule 6715
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {b^3+a^3 x^3}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (-b^3+a^3 x^3\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {b^3+a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}+\frac {2 b^3}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (-b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{3 b^2 \left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-2 b-a x^2}{3 b^2 \sqrt {b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-2 b-a x^2}{\sqrt {b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {b+a x^2}{\left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{b-2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \left (-a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 a \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 a \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {4 \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {a} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \left (\sqrt {3} a+3 \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^2} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [C] time = 1.06, size = 212, normalized size = 1.72 \begin {gather*} -\frac {(-1)^{5/6} x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (\left (3-3 i \sqrt {3}\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+4 \left ((-1)^{2/3} \Pi \left (i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\left (\sqrt [3]{-1}-1\right ) \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{2/3} \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )\right )}{3 \sqrt {\frac {i b}{a}} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.51, size = 123, normalized size = 1.00 \begin {gather*} -\frac {4 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 423, normalized size = 3.44 \begin {gather*} \left [\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} + 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 12 \, a b^{3} x + b^{4} - 4 \, \sqrt {2} {\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {\frac {1}{a b}}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {-\frac {1}{a b}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {-a b} \log \left (\frac {a^{4} x^{4} - 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {-a b}}{a^{4} x^{4} + 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}\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^{3} x^{3} + b^{3}}{{\left (a^{3} x^{3} - b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 387, normalized size = 3.15
method | result | size |
default | \(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a -2 b \right ) \left (i \underline {\hspace {1.25 ex}}\alpha a +i b +b \right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha a -i b +b}{b}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha a +b \right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{3 a}+\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{3 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}\) | \(387\) |
elliptic | \(\frac {i b \sqrt {-\frac {i x a}{b}+1}\, \sqrt {2}\, \sqrt {\frac {i x a}{b}+1}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}-\frac {2 a \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, -\frac {-\underline {\hspace {1.25 ex}}\alpha a +i b -b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}-\frac {2 \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, -\frac {-\underline {\hspace {1.25 ex}}\alpha a +i b -b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}-\frac {4 \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, -\frac {-\underline {\hspace {1.25 ex}}\alpha a +i b -b}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 a \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}+\frac {2 i \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, -\frac {-\underline {\hspace {1.25 ex}}\alpha a +i b -b}{b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}+\frac {4 i \sqrt {2}\, \sqrt {-\frac {i \left (a x +i b \right )}{b}}\, \sqrt {-\frac {i \left (-a x +i b \right )}{b}}\, \sqrt {\frac {i x a}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-\frac {i \left (a x +i b \right )}{b}}, -\frac {-\underline {\hspace {1.25 ex}}\alpha a +i b -b}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 a \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}+\frac {2 i b^{2} \sqrt {-\frac {i x a}{b}+1}\, \sqrt {2}\, \sqrt {\frac {i x a}{b}+1}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}-b \right )}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}-b \right )}\) | \(875\) |
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^{3} x^{3} + b^{3}}{{\left (a^{3} x^{3} - b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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 {\left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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