Optimal. Leaf size=218 \[ -\frac {x^{3/2}}{2 c \left (b+c x^2\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}-\frac {3 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}} \]
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Rubi [A]
time = 0.11, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1598, 294,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}-\frac {3 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 294
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{13/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^{5/2}}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}+\frac {3 \int \frac {\sqrt {x}}{b+c x^2} \, dx}{4 c}\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}+\frac {3 \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{2 c}\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2}}\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}+\frac {3 \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 c^2}+\frac {3 \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 c^2}+\frac {3 \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} \sqrt [4]{b} c^{7/4}}+\frac {3 \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} \sqrt [4]{b} c^{7/4}}\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}+\frac {3 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}-\frac {3 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \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} \sqrt [4]{b} c^{7/4}}-\frac {3 \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} \sqrt [4]{b} c^{7/4}}\\ &=-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b} c^{7/4}}+\frac {3 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}-\frac {3 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{b} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 128, normalized size = 0.59 \begin {gather*} \frac {-\frac {4 c^{3/4} x^{3/2}}{b+c x^2}-\frac {3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{b}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt [4]{b}}}{8 c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 124, normalized size = 0.57
method | result | size |
derivativedivides | \(-\frac {x^{\frac {3}{2}}}{2 c \left (c \,x^{2}+b \right )}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(124\) |
default | \(-\frac {x^{\frac {3}{2}}}{2 c \left (c \,x^{2}+b \right )}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 195, normalized size = 0.89 \begin {gather*} -\frac {x^{\frac {3}{2}}}{2 \, {\left (c^{2} x^{2} + b c\right )}} + \frac {3 \, {\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 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 185, normalized size = 0.85 \begin {gather*} -\frac {12 \, {\left (c^{2} x^{2} + b c\right )} \left (-\frac {1}{b c^{7}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {-b c^{3} \sqrt {-\frac {1}{b c^{7}}} + x} c^{2} \left (-\frac {1}{b c^{7}}\right )^{\frac {1}{4}} - c^{2} \sqrt {x} \left (-\frac {1}{b c^{7}}\right )^{\frac {1}{4}}\right ) - 3 \, {\left (c^{2} x^{2} + b c\right )} \left (-\frac {1}{b c^{7}}\right )^{\frac {1}{4}} \log \left (b c^{5} \left (-\frac {1}{b c^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + 3 \, {\left (c^{2} x^{2} + b c\right )} \left (-\frac {1}{b c^{7}}\right )^{\frac {1}{4}} \log \left (-b c^{5} \left (-\frac {1}{b c^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + 4 \, x^{\frac {3}{2}}}{8 \, {\left (c^{2} x^{2} + b c\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 = 5.00, size = 199, normalized size = 0.91 \begin {gather*} -\frac {x^{\frac {3}{2}}}{2 \, {\left (c x^{2} + b\right )} c} + \frac {3 \, \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 c^{4}} + \frac {3 \, \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 c^{4}} - \frac {3 \, \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 c^{4}} + \frac {3 \, \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 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 64, normalized size = 0.29 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}\,c^{7/4}}-\frac {x^{3/2}}{2\,c\,\left (c\,x^2+b\right )}-\frac {3\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}\,c^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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