Optimal. Leaf size=129 \[ \frac {x^{3/2}}{2 a \left (a+b x^2\right )}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}}{\sqrt {\frac {a}{b}}-x}\right )-\log \left (\frac {\sqrt {\frac {a}{b}}+\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}+x}{\sqrt {a+b x^2}}\right )}{4 \sqrt {2} a \sqrt [4]{\frac {a}{b}} b} \]
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Rubi [A]
time = 0.10, antiderivative size = 218, normalized size of antiderivative = 1.69, number of steps
used = 11, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {296, 335, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}+\frac {\int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a}\\ &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {b}}\\ &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}\\ &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}+\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}\\ &=\frac {x^{3/2}}{2 a \left (a+b x^2\right )}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.22, size = 128, normalized size = 0.99 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} x^{3/2}}{a+b x^2}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}}{8 a^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 127, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}}}{2 a \left (x^{2} b +a \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(127\) |
default | \(\frac {x^{\frac {3}{2}}}{2 a \left (x^{2} b +a \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 194, normalized size = 1.50 \begin {gather*} \frac {x^{\frac {3}{2}}}{2 \, {\left (a b x^{2} + a^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 182, normalized size = 1.41 \begin {gather*} -\frac {4 \, {\left (a b x^{2} + a^{2}\right )} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {-a^{3} b \sqrt {-\frac {1}{a^{5} b^{3}}} + x} a b \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{4}} - a b \sqrt {x} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{4}}\right ) - {\left (a b x^{2} + a^{2}\right )} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + {\left (a b x^{2} + a^{2}\right )} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 4 \, x^{\frac {3}{2}}}{8 \, {\left (a b x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs.
\(2 (95) = 190\).
time = 16.37, size = 400, normalized size = 3.10 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {4 b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 199, normalized size = 1.54 \begin {gather*} \frac {x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{3}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 64, normalized size = 0.50 \begin {gather*} \frac {x^{3/2}}{2\,a\,\left (b\,x^2+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{4\,{\left (-a\right )}^{5/4}\,b^{3/4}}+\frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{4\,{\left (-a\right )}^{5/4}\,b^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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