Optimal. Leaf size=239 \[ -\frac {21 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}+\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.19, antiderivative size = 239, 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, 290, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}-\frac {21 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{11/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{\sqrt {x} \left (b+c x^2\right )^3} \, dx\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \int \frac {1}{\sqrt {x} \left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}+\frac {21 \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 b^2}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}+\frac {21 \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^2}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}+\frac {21 \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{5/2}}+\frac {21 \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{5/2}}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}+\frac {21 \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 )}{64 b^{5/2} \sqrt {c}}+\frac {21 \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 )}{64 b^{5/2} \sqrt {c}}-\frac {21 \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 )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {21 \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 )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}-\frac {21 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {21 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}\\ &=\frac {\sqrt {x}}{4 b \left (b+c x^2\right )^2}+\frac {7 \sqrt {x}}{16 b^2 \left (b+c x^2\right )}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {21 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{11/4} \sqrt [4]{c}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 220, normalized size = 0.92 \begin {gather*} \frac {\frac {32 b^{7/4} \sqrt {x}}{\left (b+c x^2\right )^2}+\frac {56 b^{3/4} \sqrt {x}}{b+c x^2}-\frac {21 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{\sqrt [4]{c}}+\frac {21 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{\sqrt [4]{c}}-\frac {42 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{c}}+\frac {42 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{c}}}{128 b^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 149, normalized size = 0.62 \begin {gather*} -\frac {21 \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 )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {21 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {\sqrt {x} \left (11 b+7 c x^2\right )}{16 b^2 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 241, normalized size = 1.01 \begin {gather*} \frac {84 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{11} c}\right )^{\frac {1}{4}} \arctan \left (\sqrt {b^{6} \sqrt {-\frac {1}{b^{11} c}} + x} b^{8} c \left (-\frac {1}{b^{11} c}\right )^{\frac {3}{4}} - b^{8} c \sqrt {x} \left (-\frac {1}{b^{11} c}\right )^{\frac {3}{4}}\right ) + 21 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{11} c}\right )^{\frac {1}{4}} \log \left (b^{3} \left (-\frac {1}{b^{11} c}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 21 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{11} c}\right )^{\frac {1}{4}} \log \left (-b^{3} \left (-\frac {1}{b^{11} c}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (7 \, c x^{2} + 11 \, b\right )} \sqrt {x}}{64 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 209, normalized size = 0.87 \begin {gather*} \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{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 )}{64 \, b^{3} c} + \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{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 )}{64 \, b^{3} c} + \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{3} c} - \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{3} c} + \frac {7 \, c x^{\frac {5}{2}} + 11 \, b \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 0.69 \begin {gather*} \frac {\sqrt {x}}{4 \left (c \,x^{2}+b \right )^{2} b}+\frac {7 \sqrt {x}}{16 \left (c \,x^{2}+b \right ) b^{2}}+\frac {21 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 b^{3}}+\frac {21 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 b^{3}}+\frac {21 \left (\frac {b}{c}\right )^{\frac {1}{4}} \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 )}{128 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 217, normalized size = 0.91 \begin {gather*} \frac {7 \, c x^{\frac {5}{2}} + 11 \, b \sqrt {x}}{16 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )}} + \frac {21 \, {\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 {b} \sqrt {\sqrt {b} \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 {b} \sqrt {\sqrt {b} \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 {3}{4}} c^{\frac {1}{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 {3}{4}} c^{\frac {1}{4}}}\right )}}{128 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.29, size = 86, normalized size = 0.36 \begin {gather*} \frac {\frac {11\,\sqrt {x}}{16\,b}+\frac {7\,c\,x^{5/2}}{16\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}-\frac {21\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{11/4}\,c^{1/4}}-\frac {21\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{11/4}\,c^{1/4}} \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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