Optimal. Leaf size=264 \[ \frac {117 c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {117 c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{17/4}}+\frac {117 c}{16 b^4 \sqrt {x}}-\frac {117}{80 b^3 x^{5/2}}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.23, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {117 c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {117 c^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{17/4}}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {117 c}{16 b^4 \sqrt {x}}-\frac {117}{80 b^3 x^{5/2}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x^{7/2} \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13 \int \frac {1}{x^{7/2} \left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {117 \int \frac {1}{x^{7/2} \left (b+c x^2\right )} \, dx}{32 b^2}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}-\frac {(117 c) \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 b^4}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {\left (117 c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^4}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}-\frac {\left (117 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4}+\frac {\left (117 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {(117 c) \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^4}+\frac {(117 c) \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^4}+\frac {\left (117 c^{5/4}\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 )}{64 \sqrt {2} b^{17/4}}+\frac {\left (117 c^{5/4}\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 )}{64 \sqrt {2} b^{17/4}}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac {117 c^{5/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {\left (117 c^{5/4}\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 )}{32 \sqrt {2} b^{17/4}}-\frac {\left (117 c^{5/4}\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 )}{32 \sqrt {2} b^{17/4}}\\ &=-\frac {117}{80 b^3 x^{5/2}}+\frac {117 c}{16 b^4 \sqrt {x}}+\frac {1}{4 b x^{5/2} \left (b+c x^2\right )^2}+\frac {13}{16 b^2 x^{5/2} \left (b+c x^2\right )}-\frac {117 c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {117 c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {117 c^{5/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.11 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {5}{4},3;-\frac {1}{4};-\frac {c x^2}{b}\right )}{5 b^3 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 171, normalized size = 0.65 \begin {gather*} -\frac {117 c^{5/4} \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 )}{32 \sqrt {2} b^{17/4}}-\frac {117 c^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{17/4}}+\frac {-32 b^3+416 b^2 c x^2+1053 b c^2 x^4+585 c^3 x^6}{80 b^4 x^{5/2} \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 306, normalized size = 1.16 \begin {gather*} -\frac {2340 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {1}{4}} \arctan \left (-\frac {1601613 \, b^{4} c^{4} \sqrt {x} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {1}{4}} - \sqrt {-2565164201769 \, b^{9} c^{5} \sqrt {-\frac {c^{5}}{b^{17}}} + 2565164201769 \, c^{8} x} b^{4} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {1}{4}}}{1601613 \, c^{5}}\right ) - 585 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (1601613 \, b^{13} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {3}{4}} + 1601613 \, c^{4} \sqrt {x}\right ) + 585 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (-1601613 \, b^{13} \left (-\frac {c^{5}}{b^{17}}\right )^{\frac {3}{4}} + 1601613 \, c^{4} \sqrt {x}\right ) - 4 \, {\left (585 \, c^{3} x^{6} + 1053 \, b c^{2} x^{4} + 416 \, b^{2} c x^{2} - 32 \, b^{3}\right )} \sqrt {x}}{320 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 232, normalized size = 0.88 \begin {gather*} \frac {117 \, \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 )}{64 \, b^{5} c} + \frac {117 \, \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 )}{64 \, b^{5} c} - \frac {117 \, \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 )}{128 \, b^{5} c} + \frac {117 \, \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 )}{128 \, b^{5} c} + \frac {21 \, c^{3} x^{\frac {7}{2}} + 25 \, b c^{2} x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} + \frac {2 \, {\left (15 \, c x^{2} - b\right )}}{5 \, b^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 192, normalized size = 0.73 \begin {gather*} \frac {21 c^{3} x^{\frac {7}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{4}}+\frac {25 c^{2} x^{\frac {3}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{3}}+\frac {117 \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}+\frac {117 \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}+\frac {117 \sqrt {2}\, c \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 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}+\frac {6 c}{b^{4} \sqrt {x}}-\frac {2}{5 b^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.14, size = 243, normalized size = 0.92 \begin {gather*} \frac {585 \, c^{3} x^{6} + 1053 \, b c^{2} x^{4} + 416 \, b^{2} c x^{2} - 32 \, b^{3}}{80 \, {\left (b^{4} c^{2} x^{\frac {13}{2}} + 2 \, b^{5} c x^{\frac {9}{2}} + b^{6} x^{\frac {5}{2}}\right )}} + \frac {117 \, c^{2} {\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 )}}{128 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 109, normalized size = 0.41 \begin {gather*} \frac {\frac {26\,c\,x^2}{5\,b^2}-\frac {2}{5\,b}+\frac {1053\,c^2\,x^4}{80\,b^3}+\frac {117\,c^3\,x^6}{16\,b^4}}{b^2\,x^{5/2}+c^2\,x^{13/2}+2\,b\,c\,x^{9/2}}-\frac {117\,{\left (-c\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{17/4}}+\frac {117\,{\left (-c\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{17/4}} \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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