Optimal. Leaf size=230 \[ -\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}+\frac {5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1598, 296,
331, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {5 \sqrt [4]{c} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x^{3/2} \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}+\frac {5 \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{4 b}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}-\frac {(5 c) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{4 b^2}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}-\frac {(5 c) \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{2 b^2}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}+\frac {\left (5 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^2}-\frac {\left (5 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}-\frac {5 \text {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^2}-\frac {5 \text {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^2}-\frac {\left (5 \sqrt [4]{c}\right ) \text {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^{9/4}}-\frac {\left (5 \sqrt [4]{c}\right ) \text {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^{9/4}}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}-\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}-\frac {\left (5 \sqrt [4]{c}\right ) \text {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^{9/4}}+\frac {\left (5 \sqrt [4]{c}\right ) \text {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^{9/4}}\\ &=-\frac {5}{2 b^2 \sqrt {x}}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}+\frac {5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 138, normalized size = 0.60 \begin {gather*} \frac {-\frac {4 \sqrt [4]{b} \left (4 b+5 c x^2\right )}{\sqrt {x} \left (b+c x^2\right )}+5 \sqrt {2} \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{8 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 136, normalized size = 0.59
method | result | size |
derivativedivides | \(-\frac {2}{b^{2} \sqrt {x}}-\frac {2 c \left (\frac {x^{\frac {3}{2}}}{4 c \,x^{2}+4 b}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}}\) | \(136\) |
default | \(-\frac {2}{b^{2} \sqrt {x}}-\frac {2 c \left (\frac {x^{\frac {3}{2}}}{4 c \,x^{2}+4 b}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}}\) | \(136\) |
risch | \(-\frac {2}{b^{2} \sqrt {x}}-\frac {c \,x^{\frac {3}{2}}}{2 b^{2} \left (c \,x^{2}+b \right )}-\frac {5 \sqrt {2}\, \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{16 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 208, normalized size = 0.90 \begin {gather*} -\frac {5 \, c x^{2} + 4 \, b}{2 \, {\left (b^{2} c x^{\frac {5}{2}} + b^{3} \sqrt {x}\right )}} - \frac {5 \, 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 )}}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 208, normalized size = 0.90 \begin {gather*} \frac {20 \, {\left (b^{2} c x^{3} + b^{3} x\right )} \left (-\frac {c}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {125 \, b^{2} c \sqrt {x} \left (-\frac {c}{b^{9}}\right )^{\frac {1}{4}} - \sqrt {-15625 \, b^{5} c \sqrt {-\frac {c}{b^{9}}} + 15625 \, c^{2} x} b^{2} \left (-\frac {c}{b^{9}}\right )^{\frac {1}{4}}}{125 \, c}\right ) - 5 \, {\left (b^{2} c x^{3} + b^{3} x\right )} \left (-\frac {c}{b^{9}}\right )^{\frac {1}{4}} \log \left (125 \, b^{7} \left (-\frac {c}{b^{9}}\right )^{\frac {3}{4}} + 125 \, c \sqrt {x}\right ) + 5 \, {\left (b^{2} c x^{3} + b^{3} x\right )} \left (-\frac {c}{b^{9}}\right )^{\frac {1}{4}} \log \left (-125 \, b^{7} \left (-\frac {c}{b^{9}}\right )^{\frac {3}{4}} + 125 \, c \sqrt {x}\right ) - 4 \, {\left (5 \, c x^{2} + 4 \, b\right )} \sqrt {x}}{8 \, {\left (b^{2} c x^{3} + b^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 4.45, size = 210, normalized size = 0.91 \begin {gather*} -\frac {5 \, c x^{2} + 4 \, b}{2 \, {\left (c x^{\frac {5}{2}} + b \sqrt {x}\right )} b^{2}} - \frac {5 \, \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 )}{8 \, b^{3} c^{2}} - \frac {5 \, \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 )}{8 \, b^{3} c^{2}} + \frac {5 \, \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 )}{16 \, b^{3} c^{2}} - \frac {5 \, \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 )}{16 \, b^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 77, normalized size = 0.33 \begin {gather*} \frac {5\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{9/4}}-\frac {5\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{4\,b^{9/4}}-\frac {\frac {2}{b}+\frac {5\,c\,x^2}{2\,b^2}}{b\,\sqrt {x}+c\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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