Optimal. Leaf size=243 \[ -\frac {11 c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}-\frac {11 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c}{6 b^3 x^{3/2}}-\frac {11}{14 b^2 x^{7/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )} \]
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Rubi [A] time = 0.21, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {11 c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}-\frac {11 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c}{6 b^3 x^{3/2}}-\frac {11}{14 b^2 x^{7/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 290
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x^{9/2} \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}+\frac {11 \int \frac {1}{x^{9/2} \left (b+c x^2\right )} \, dx}{4 b}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}-\frac {(11 c) \int \frac {1}{x^{5/2} \left (b+c x^2\right )} \, dx}{4 b^2}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}+\frac {\left (11 c^2\right ) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{4 b^3}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}+\frac {\left (11 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{2 b^3}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}+\frac {\left (11 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^{7/2}}+\frac {\left (11 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^{7/2}}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}+\frac {\left (11 c^{3/2}\right ) \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 )}{8 b^{7/2}}+\frac {\left (11 c^{3/2}\right ) \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 )}{8 b^{7/2}}-\frac {\left (11 c^{7/4}\right ) \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 )}{8 \sqrt {2} b^{15/4}}-\frac {\left (11 c^{7/4}\right ) \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 )}{8 \sqrt {2} b^{15/4}}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}-\frac {11 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}+\frac {\left (11 c^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}-\frac {\left (11 c^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}\\ &=-\frac {11}{14 b^2 x^{7/2}}+\frac {11 c}{6 b^3 x^{3/2}}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}-\frac {11 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}-\frac {11 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{15/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.12 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {7}{4},2;-\frac {3}{4};-\frac {c x^2}{b}\right )}{7 b^2 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 160, normalized size = 0.66 \begin {gather*} -\frac {11 c^{7/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {x}}\right )}{4 \sqrt {2} b^{15/4}}+\frac {11 c^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{4 \sqrt {2} b^{15/4}}+\frac {-12 b^2+44 b c x^2+77 c^2 x^4}{42 b^3 x^{7/2} \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 245, normalized size = 1.01 \begin {gather*} \frac {924 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{11} c^{2} \sqrt {x} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {3}{4}} - \sqrt {b^{8} \sqrt {-\frac {c^{7}}{b^{15}}} + c^{4} x} b^{11} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {3}{4}}}{c^{7}}\right ) + 231 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {1}{4}} \log \left (11 \, b^{4} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {1}{4}} + 11 \, c^{2} \sqrt {x}\right ) - 231 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {1}{4}} \log \left (-11 \, b^{4} \left (-\frac {c^{7}}{b^{15}}\right )^{\frac {1}{4}} + 11 \, c^{2} \sqrt {x}\right ) + 4 \, {\left (77 \, c^{2} x^{4} + 44 \, b c x^{2} - 12 \, b^{2}\right )} \sqrt {x}}{168 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 212, normalized size = 0.87 \begin {gather*} \frac {11 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \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 )}{8 \, b^{4}} + \frac {11 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \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 )}{8 \, b^{4}} + \frac {11 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{4}} - \frac {11 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{4}} + \frac {c^{2} \sqrt {x}}{2 \, {\left (c x^{2} + b\right )} b^{3}} + \frac {2 \, {\left (14 \, c x^{2} - 3 \, b\right )}}{21 \, b^{3} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 178, normalized size = 0.73 \begin {gather*} \frac {c^{2} \sqrt {x}}{2 \left (c \,x^{2}+b \right ) b^{3}}+\frac {11 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 b^{4}}+\frac {11 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 b^{4}}+\frac {11 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, c^{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 )}{16 b^{4}}+\frac {4 c}{3 b^{3} x^{\frac {3}{2}}}-\frac {2}{7 b^{2} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.16, size = 224, normalized size = 0.92 \begin {gather*} \frac {77 \, c^{2} x^{4} + 44 \, b c x^{2} - 12 \, b^{2}}{42 \, {\left (b^{3} c x^{\frac {11}{2}} + b^{4} x^{\frac {7}{2}}\right )}} + \frac {11 \, {\left (\frac {2 \, \sqrt {2} c^{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 {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} c^{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 {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}}\right )}}{16 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 87, normalized size = 0.36 \begin {gather*} \frac {\frac {22\,c\,x^2}{21\,b^2}-\frac {2}{7\,b}+\frac {11\,c^2\,x^4}{6\,b^3}}{b\,x^{7/2}+c\,x^{11/2}}+\frac {11\,{\left (-c\right )}^{7/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{15/4}}+\frac {11\,{\left (-c\right )}^{7/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{15/4}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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