Optimal. Leaf size=227 \[ \frac {2}{3} \sqrt {b^2+a^2 x^3}+\frac {\left (-(-1)^{3/4} a \sqrt {a-i b} b+\sqrt [4]{-1} \sqrt {a-i b} c\right ) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 a \sqrt {b}}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\left (-(-1)^{3/4} a \sqrt {a+i b} b-\sqrt [4]{-1} \sqrt {a+i b} c\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {b^2+a^2 x^3}}{\sqrt {2} \sqrt {a+i b} \sqrt {b}}\right )}{3 a \sqrt {b}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(609\) vs. \(2(227)=454\).
time = 2.15, antiderivative size = 609, normalized size of antiderivative = 2.68, number of steps
used = 18, number of rules used = 13, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.260, Rules used = {6857, 272,
52, 65, 214, 6847, 839, 841, 1183, 648, 632, 212, 642} \begin {gather*} \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 )-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}} \end {gather*}
Antiderivative was successfully verified.
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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 \left (b^2+a^2 x^6\right )} \, dx &=\int \left (\frac {2 \sqrt {b^2+a^2 x^3}}{x}-\frac {x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2+a^2 x^6}\right ) \, dx\\ &=2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx-\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\\ &=-\left (\frac {1}{3} \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 )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\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 a^2}+\frac {1}{3} \left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\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 a^2}+\frac {\left (4 b^2\right ) \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 a^2}\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )-\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^4 b^{3/2} \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^4 b^{3/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\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 \sqrt {a^2+b^2}}+\frac {\left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\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 \sqrt {a^2+b^2}}-\frac {\left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {\left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\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 \sqrt {a^2+b^2}}-\frac {\left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\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 \sqrt {a^2+b^2}}\\ &=\frac {2}{3} \sqrt {b^2+a^2 x^3}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 189, normalized size = 0.83 \begin {gather*} \frac {1}{3} \left (2 \sqrt {b^2+a^2 x^3}+\frac {\sqrt [4]{-1} \sqrt {a-i b} (-i a b+c) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{a \sqrt {b}}-\frac {\sqrt [4]{-1} \sqrt {a+i 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 \sqrt {b}}-4 b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )\right ) \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 = 0.32, size = 703, normalized size = 3.10
method | result | size |
default | \(\frac {2 \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}}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{6}+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 )}{6 a^{3} b^{2}}\) | \(703\) |
elliptic | \(\text {Expression too large to display}\) | \(12449\) |
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. 9711 vs.
\(2 (163) = 326\).
time = 5.27, size = 9711, normalized size = 42.78 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{3} + b^{2}} \left (a^{2} x^{6} + 2 b^{2} + c x^{3}\right )}{x \left (a^{2} x^{6} + b^{2}\right )}\, dx \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 = 13.68, size = 246, normalized size = 1.08 \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 )\,{\left (b-\sqrt {a^2\,x^3+b^2}\right )}^3}{x^6}\right )}{3}+\frac {\sqrt {\frac {1}{36}{}\mathrm {i}}\,\ln \left (\frac {2\,{\left (-1\right )}^{1/4}\,b^2+{\left (-1\right )}^{1/4}\,a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b\,1{}\mathrm {i}}-{\left (-1\right )}^{3/4}\,a\,b}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\sqrt {a+b\,1{}\mathrm {i}}\,\left (c+a\,b\,1{}\mathrm {i}\right )}{a\,\sqrt {b}}+\frac {\sqrt {\frac {1}{36}{}\mathrm {i}}\,\ln \left (\frac {2\,{\left (-1\right )}^{1/4}\,b^2+{\left (-1\right )}^{1/4}\,a^2\,x^3+{\left (-1\right )}^{3/4}\,a\,b+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-a+b\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\sqrt {-a+b\,1{}\mathrm {i}}\,\left (c-a\,b\,1{}\mathrm {i}\right )}{a\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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