Optimal. Leaf size=215 \[ -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}-\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} c^{9/4}}-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c} \]
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Rubi [A] time = 0.19, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1584, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}-\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} c^{9/4}}-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{11/2}}{b x^2+c x^4} \, dx &=\int \frac {x^{7/2}}{b+c x^2} \, dx\\ &=\frac {2 x^{5/2}}{5 c}-\frac {b \int \frac {x^{3/2}}{b+c x^2} \, dx}{c}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}+\frac {b^2 \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{c^2}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}+\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}+\frac {b^{3/2} \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 c^{5/2}}+\frac {b^{3/2} \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 c^{5/2}}-\frac {b^{5/4} \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} c^{9/4}}-\frac {b^{5/4} \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} c^{9/4}}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}-\frac {b^{5/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {b^{5/4} \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} c^{9/4}}-\frac {b^{5/4} \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} c^{9/4}}\\ &=-\frac {2 b \sqrt {x}}{c^2}+\frac {2 x^{5/2}}{5 c}-\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}-\frac {b^{5/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {b^{5/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 203, normalized size = 0.94 \[ \frac {-5 \sqrt {2} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )+5 \sqrt {2} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )-10 \sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )+10 \sqrt {2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )-40 b \sqrt [4]{c} \sqrt {x}+8 c^{5/4} x^{5/2}}{20 c^{9/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 170, normalized size = 0.79 \[ \frac {20 \, c^{2} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b c^{7} \sqrt {x} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {3}{4}} - \sqrt {c^{4} \sqrt {-\frac {b^{5}}{c^{9}}} + b^{2} x} c^{7} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {3}{4}}}{b^{5}}\right ) + 5 \, c^{2} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {1}{4}} \log \left (c^{2} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {1}{4}} + b \sqrt {x}\right ) - 5 \, c^{2} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {1}{4}} \log \left (-c^{2} \left (-\frac {b^{5}}{c^{9}}\right )^{\frac {1}{4}} + b \sqrt {x}\right ) + 4 \, {\left (c x^{2} - 5 \, b\right )} \sqrt {x}}{10 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 196, normalized size = 0.91 \[ \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{3}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{3}} + \frac {2 \, {\left (c^{4} x^{\frac {5}{2}} - 5 \, b c^{3} \sqrt {x}\right )}}{5 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 152, normalized size = 0.71 \[ \frac {2 x^{\frac {5}{2}}}{5 c}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{2 c^{2}}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{2 c^{2}}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, b \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{4 c^{2}}-\frac {2 b \sqrt {x}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 194, normalized size = 0.90 \[ \frac {2 \, {\left (c x^{\frac {5}{2}} - 5 \, b \sqrt {x}\right )}}{5 \, c^{2}} + \frac {\frac {2 \, \sqrt {2} b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} b^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} b^{\frac {5}{4}} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} b^{\frac {5}{4}} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{c^{\frac {1}{4}}}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 67, normalized size = 0.31 \[ \frac {2\,x^{5/2}}{5\,c}-\frac {2\,b\,\sqrt {x}}{c^2}-\frac {{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{c^{9/4}}+\frac {{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,1{}\mathrm {i}}{c^{9/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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