Optimal. Leaf size=245 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}} \]
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Rubi [C] time = 8.42, antiderivative size = 956, normalized size of antiderivative = 3.90, number of steps used = 185, number of rules used = 21, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.477, Rules used = {2056, 1586, 6715, 6725, 1729, 1209, 1198, 220, 1196, 1211, 1699, 208, 1248, 735, 844, 217, 206, 725, 205, 1217, 1707} \begin {gather*} -\frac {2 \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 {a^2 x^3+b^2 x}}-\frac {2 \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 {a^2 x^3+b^2 x}}-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\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 {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (3+i \sqrt {3}\right ) \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 )}{6 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \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 {a^2 x^3+b^2 x}}+\frac {\left (3-i \sqrt {3}\right ) \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 )}{6 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \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 {a^2 x^3+b^2 x}}-\frac {\left (i-\sqrt {3}\right ) \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 \left (3 i-\sqrt {3}\right ) \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\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 )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 208
Rule 217
Rule 220
Rule 725
Rule 735
Rule 844
Rule 1196
Rule 1198
Rule 1209
Rule 1211
Rule 1217
Rule 1248
Rule 1586
Rule 1699
Rule 1707
Rule 1729
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (-b^6+a^6 x^6\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )}{\sqrt {x} \left (-b^6+a^6 x^6\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 {\sqrt {b^2+a^2 x^4} \left (b^4-a^2 b^2 x^4+a^4 x^8\right )}{-b^6+a^6 x^{12}} \, 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 {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}-\sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}-i \sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}+i \sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}+\sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt {a} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
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Mathematica [C] time = 2.62, size = 296, normalized size = 1.21 \begin {gather*} \frac {2 i x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (-3 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\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 = 0.61, size = 245, normalized size = 1.00 \begin {gather*} -\frac {2 \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 {\tan ^{-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 {2} \sqrt {a} \sqrt {b}}-\frac {2 \tanh ^{-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 {\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 {2} \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 828, normalized size = 3.38 \begin {gather*} \left [-\frac {2 \, \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 ) - \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 ) - 4 \, \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 )}{24 \, a b}, \frac {2 \, \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 ) + \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 ) - 4 \, \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 )}{24 \, 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^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\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 = 0.05, size = 681, normalized size = 2.78 \begin {gather*} \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 {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 {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 )}+\frac {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 {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 )} \end {gather*}
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^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\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 = -0.00 \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^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\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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