Optimal. Leaf size=149 \[ \frac {\sqrt {b x+a x^3}}{2 a b \left (b+a x^2\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 187, normalized size of antiderivative = 1.26, number of steps
used = 9, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2081, 1268,
477, 482, 21, 413, 218, 212, 209} \begin {gather*} -\frac {\sqrt {x} \sqrt {a x^2+b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {a x^3+b x}}-\frac {\sqrt {x} \sqrt {a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {a x^3+b x}}+\frac {x}{2 a b \sqrt {a x^3+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 209
Rule 212
Rule 218
Rule 413
Rule 477
Rule 482
Rule 1268
Rule 2081
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {b x+a x^3} \left (-b^2+a^2 x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {x^{3/2}}{\sqrt {b+a x^2} \left (-b^2+a^2 x^4\right )} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {x^{3/2}}{\left (-b+a x^2\right ) \left (b+a x^2\right )^{3/2}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-b+a x^4\right ) \left (b+a x^4\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}\\ &=\frac {x}{2 a b \sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {-b-a x^4}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 a b \sqrt {b x+a x^3}}\\ &=\frac {x}{2 a b \sqrt {b x+a x^3}}+\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx,x,\sqrt {x}\right )}{2 a b \sqrt {b x+a x^3}}\\ &=\frac {x}{2 a b \sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-4 a b x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt {b+a x^2}}\right )}{2 a b \sqrt {b x+a x^3}}\\ &=\frac {x}{2 a b \sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 \sqrt {a} \sqrt {b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b+a x^2}}\right )}{4 a b \sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+2 \sqrt {a} \sqrt {b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b+a x^2}}\right )}{4 a b \sqrt {b x+a x^3}}\\ &=\frac {x}{2 a b \sqrt {b x+a x^3}}-\frac {\sqrt {x} \sqrt {b+a x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {b x+a x^3}}-\frac {\sqrt {x} \sqrt {b+a x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4} \sqrt {b x+a x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 152, normalized size = 1.02 \begin {gather*} \frac {\sqrt {x} \left (4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}-\sqrt {2} \sqrt {b+a x^2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )-\sqrt {2} \sqrt {b+a x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{8 a^{5/4} b^{5/4} \sqrt {x \left (b+a x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.10, size = 448, normalized size = 3.01
method | result | size |
elliptic | \(\frac {x}{2 a b \sqrt {\left (x^{2}+\frac {b}{a}\right ) a x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{4 a^{2} b \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{4 a^{2} \sqrt {a b}\, \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\) | \(421\) |
default | \(\frac {\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}}{2 a}+\frac {\frac {x}{b \sqrt {\left (x^{2}+\frac {b}{a}\right ) a x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 b a \sqrt {a \,x^{3}+b x}}}{2 a}\) | \(448\) |
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. 426 vs.
\(2 (113) = 226\).
time = 0.45, size = 426, normalized size = 2.86 \begin {gather*} -\frac {4 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{3} + b x} a^{4} b^{4} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {3}{4}}}{a x^{2} + b}\right ) + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} + 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{5} b^{4} x^{2} + a^{4} b^{5}\right )} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {3}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{4} b^{3} x^{3} + a^{3} b^{4} x\right )} \sqrt {\frac {1}{a^{5} b^{5}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 8 \, {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{5} b^{4} x^{2} + a^{4} b^{5}\right )} \left (\frac {1}{a^{5} b^{5}}\right )^{\frac {3}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{4} b^{3} x^{3} + a^{3} b^{4} x\right )} \sqrt {\frac {1}{a^{5} b^{5}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) - 8 \, \sqrt {a x^{3} + b x}}{16 \, {\left (a^{2} b x^{2} + a b^{2}\right )}} \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 {x^{2}}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} - b\right ) \left (a x^{2} + b\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 = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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