Optimal. Leaf size=247 \[ \frac {\sqrt {b^2+a^2 x^3} \left (-2 b^2-a^2 x^3-2 c x^3\right )}{6 b^2 x^6}+\frac {\sqrt {a-i b} \left ((-1)^{3/4} a^2 b-\sqrt [4]{-1} a c\right ) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+i b} \left (-\sqrt [4]{-1} a^2 b+(-1)^{3/4} a c\right ) \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (a^4+4 a^2 b^2-2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(744\) vs. \(2(247)=494\).
time = 2.53, antiderivative size = 744, normalized size of antiderivative = 3.01, number of steps
used = 27, number of rules used = 15, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6857, 272,
43, 44, 65, 214, 52, 6847, 839, 841, 1183, 648, 632, 212, 642} \begin {gather*} -\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\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 )}{3 b}-\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}+\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (-\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}-\sqrt {2} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a^2 x^3+b^2}+\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\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 43
Rule 44
Rule 52
Rule 65
Rule 212
Rule 214
Rule 272
Rule 632
Rule 642
Rule 648
Rule 839
Rule 841
Rule 1183
Rule 6847
Rule 6857
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 {a^2 \sqrt {b^2+a^2 x^3}}{b^2 x}+\frac {a^2 x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2 \left (b^2+a^2 x^6\right )}\right ) \, dx\\ &=2 \int \frac {\sqrt {b^2+a^2 x^3}}{x^7} \, dx-\frac {a^2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx}{b^2}+\frac {a^2 \int \frac {x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2+a^2 x^6} \, dx}{b^2}+\frac {c \int \frac {\sqrt {b^2+a^2 x^3}}{x^4} \, dx}{b^2}\\ &=\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^3} \, dx,x,x^3\right )-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )}{3 b^2}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x} \left (-c+a^2 x\right )}{b^2+a^2 x^2} \, dx,x,x^3\right )}{3 b^2}+\frac {c \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^2} \, dx,x,x^3\right )}{3 b^2}\\ &=-\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 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{3} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )+\frac {\text {Subst}\left (\int \frac {-a^2 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x}{\sqrt {b^2+a^2 x} \left (b^2+a^2 x^2\right )} \, dx,x,x^3\right )}{3 b^2}+\frac {\left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}\\ &=-\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 {2}{3} \text {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 {2 \text {Subst}\left (\int \frac {-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x^2}{a^4 b^2+a^2 b^4-2 a^2 b^2 x^2+a^2 x^4} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}-\frac {a^4 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{12 b^2}+\frac {c \text {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 {\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 {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \text {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 {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )-\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )+\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\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^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 )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\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^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 )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}\\ &=-\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^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 )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 224, normalized size = 0.91 \begin {gather*} \frac {-\frac {b \sqrt {b^2+a^2 x^3} \left (2 b^2+\left (a^2+2 c\right ) x^3\right )}{x^6}+2 (-1)^{3/4} a \sqrt {a-i b} \sqrt {b} (a b+i c) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )+2 \sqrt [4]{-1} a \sqrt {a+i b} \sqrt {b} (a b-i c) \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )+a^2 \left (a^2+4 b^2-2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.33, size = 864, normalized size = 3.50
method | result | size |
risch | \(-\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )}{6 b^{2} x^{6}}-\frac {a^{2} \left (-\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{6} a^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} c +a^{2} b^{2}+b^{2} c \right ) \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 (2 a^{2} \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{5}-b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-i a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4}-a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (-a \,b^{2}\right )^{\frac {2}{3}}-i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b^{2} a +i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b^{2}+\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2} a +\left (-a \,b^{2}\right )^{\frac {2}{3}} b^{2}\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 ^{5} a^{3}-i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}-2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,b^{2}+i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,b^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-i \sqrt {3}\, b^{4}+3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2}+3 b^{4}}{2 b^{2} \left (a^{2}+b^{2}\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 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{2}+b^{2}\right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{3} b^{2}}-\frac {2 \left (a^{2}+4 b^{2}-2 c \right ) \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{4 b^{2}}\) | \(746\) |
default | \(\text {Expression too large to display}\) | \(864\) |
elliptic | \(\text {Expression too large to display}\) | \(12546\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10278 vs.
\(2 (194) = 388\).
time = 4.83, size = 10278, normalized size = 41.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.50, size = 274, normalized size = 1.11 \begin {gather*} \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+4\,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 {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b\,1{}\mathrm {i}+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-b+a\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (-a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {-b+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {a\,b\,1{}\mathrm {i}+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b+a\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {b+a\,1{}\mathrm {i}}}{6\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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