Optimal. Leaf size=139 \[ \frac {\sqrt {x} \left (-3 a+b x^2\right )}{16 a b \left (a+b x^2\right )^2}+\frac {3 \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}}{\sqrt {\frac {a}{b}}-x}\right )+\log \left (\frac {\sqrt {\frac {a}{b}}+\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}+x}{\sqrt {a+b x^2}}\right )\right )}{32 \sqrt {2} a \left (\frac {a}{b}\right )^{3/4} b^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 242, normalized size of antiderivative = 1.74, number of steps
used = 12, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {294, 296, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 294
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2} \, dx}{8 b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \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 )}{64 a^{3/2} b^{3/2}}+\frac {3 \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 )}{64 a^{3/2} b^{3/2}}-\frac {3 \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 )}{64 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \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 )}{64 \sqrt {2} a^{7/4} b^{5/4}}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.29, size = 137, normalized size = 0.99 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-3 a+b x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{7/4} b^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 138, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\) | \(138\) |
default | \(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 221, normalized size = 1.59 \begin {gather*} \frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} + \frac {3 \, {\left (\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 {a} \sqrt {\sqrt {a} \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 {a} \sqrt {\sqrt {a} \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 {3}{4}} b^{\frac {1}{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 {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a b} \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. 257 vs.
\(2 (112) = 224\).
time = 0.62, size = 257, normalized size = 1.85 \begin {gather*} \frac {12 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{4} b^{2} \sqrt {-\frac {1}{a^{7} b^{5}}} + x} a^{5} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {3}{4}} - a^{5} b^{4} \sqrt {x} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {3}{4}}\right ) + 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (b x^{2} - 3 \, a\right )} \sqrt {x}}{64 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\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. 666 vs.
\(2 (112) = 224\).
time = 145.38, size = 666, normalized size = 4.79 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {12 a^{2} \sqrt {x}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {4 a b x^{\frac {5}{2}}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {12 a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 211, normalized size = 1.52 \begin {gather*} \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{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 )}{64 \, a^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{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 )}{64 \, a^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} + \frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 85, normalized size = 0.61 \begin {gather*} \frac {\frac {x^{5/2}}{16\,a}-\frac {3\,\sqrt {x}}{16\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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