Optimal. Leaf size=202 \[ -\frac {\sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{5/4}}-\frac {2}{b \sqrt {x}} \]
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Rubi [A] time = 0.16, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1584, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{5/4}}-\frac {2}{b \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{b x^2+c x^4} \, dx &=\int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx\\ &=-\frac {2}{b \sqrt {x}}-\frac {c \int \frac {\sqrt {x}}{b+c x^2} \, dx}{b}\\ &=-\frac {2}{b \sqrt {x}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {2}{b \sqrt {x}}+\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {2}{b \sqrt {x}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b}-\frac {\sqrt [4]{c} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{5/4}}\\ &=-\frac {2}{b \sqrt {x}}-\frac {\sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}\\ &=-\frac {2}{b \sqrt {x}}+\frac {\sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{5/4}}-\frac {\sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}+\frac {\sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.13 \[ -\frac {2 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {c x^2}{b}\right )}{b \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 142, normalized size = 0.70 \[ \frac {4 \, b x \left (-\frac {c}{b^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b c \sqrt {x} \left (-\frac {c}{b^{5}}\right )^{\frac {1}{4}} - \sqrt {-b^{3} c \sqrt {-\frac {c}{b^{5}}} + c^{2} x} b \left (-\frac {c}{b^{5}}\right )^{\frac {1}{4}}}{c}\right ) - b x \left (-\frac {c}{b^{5}}\right )^{\frac {1}{4}} \log \left (b^{4} \left (-\frac {c}{b^{5}}\right )^{\frac {3}{4}} + c \sqrt {x}\right ) + b x \left (-\frac {c}{b^{5}}\right )^{\frac {1}{4}} \log \left (-b^{4} \left (-\frac {c}{b^{5}}\right )^{\frac {3}{4}} + c \sqrt {x}\right ) - 4 \, \sqrt {x}}{2 \, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 190, normalized size = 0.94 \[ -\frac {2}{b \sqrt {x}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{2} c^{2}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{2} c^{2}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{2} c^{2}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 140, normalized size = 0.69 \[ -\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {b}{c}\right )^{\frac {1}{4}} b}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {b}{c}\right )^{\frac {1}{4}} b}-\frac {\sqrt {2}\, \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{4 \left (\frac {b}{c}\right )^{\frac {1}{4}} b}-\frac {2}{b \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 186, normalized size = 0.92 \[ -\frac {c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, b} - \frac {2}{b \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.53, size = 54, normalized size = 0.27 \[ \frac {{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{b^{5/4}}-\frac {{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{b^{5/4}}-\frac {2}{b\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.32, size = 170, normalized size = 0.84 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge c = 0 \\- \frac {2}{5 c x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: c = 0 \\- \frac {2}{b \sqrt {x}} + \frac {\left (-1\right )^{\frac {3}{4}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 b^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {3}{4}} \log {\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 b^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}}} - \frac {\left (-1\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{b} \sqrt [4]{\frac {1}{c}}} \right )}}{b^{\frac {5}{4}} \sqrt [4]{\frac {1}{c}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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