Optimal. Leaf size=239 \[ -\frac {2 \sqrt {2 \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}} \]
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Rubi [F]
time = 2.29, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {b^5+a^5 x^5}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (-b^5+a^5 x^5\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b^5+a^5 x^{10}}{\sqrt {b^2+a^2 x^4} \left (-b^5+a^5 x^{10}\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 ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}+\frac {2 b^5}{\sqrt {b^2+a^2 x^4} \left (-b^5+a^5 x^{10}\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 ) \text {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^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (-b^5+a^5 x^{10}\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^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{5 b^4 \left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-4 b^3-3 a b^2 x^2-2 a^2 b x^4-a^3 x^6}{5 b^4 \sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\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 ) \text {Subst}\left (\int \frac {1}{\left (b-a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-4 b^3-3 a b^2 x^2-2 a^2 b x^4-a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \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 ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {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 )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {4 b^3}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )}-\frac {3 a b^2 x^2}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )}-\frac {2 a^2 b x^4}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )}-\frac {a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \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 )}{5 \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 ) \text {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 )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \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 )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {4 \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 )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4+a b^3 x^2+a^2 b^2 x^4+a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 214, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \left (2 \sqrt {-1+\sqrt {5}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+2 \sqrt {1+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{5 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\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 = 0.24, size = 482, normalized size = 2.02
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 (a^{4} \textit {\_Z}^{4}+a^{3} b \,\textit {\_Z}^{3}+a^{2} b^{2} \textit {\_Z}^{2}+a \,b^{3} \textit {\_Z} +b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-2 b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-4 b^{3}\right ) \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a b -i b^{2}\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 \left (i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a b +b^{2}\right ) a}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{\left (4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+3 b \,\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+b^{3}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{5 b^{2}}+\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 )}{5 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}\) | \(482\) |
elliptic | \(\text {Expression too large to display}\) | \(2129\) |
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. 997 vs.
\(2 (177) = 354\).
time = 0.60, size = 2075, normalized size = 8.68 \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 {\left (a x + b\right ) \left (a^{4} x^{4} - a^{3} b x^{3} + a^{2} b^{2} x^{2} - a b^{3} x + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a^{4} x^{4} + a^{3} b x^{3} + a^{2} b^{2} x^{2} + a b^{3} x + b^{4}\right )}\, dx \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*} \text {could not integrate} \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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