Optimal. Leaf size=239 \[ \frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}-\frac {5 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}} \]
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Rubi [A]
time = 0.12, 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 = {1598, 296,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {5 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}-\frac {5 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}+\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 296
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 )^3} \, dx &=\int \frac {\sqrt {x}}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 \int \frac {\sqrt {x}}{\left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}+\frac {5 \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 b^2}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}+\frac {5 \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^2}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^2 \sqrt {c}}+\frac {5 \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^2 \sqrt {c}}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \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 )}{64 b^2 c}+\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 )}{64 b^2 c}+\frac {5 \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 )}{64 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \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 )}{64 \sqrt {2} b^{9/4} c^{3/4}}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}+\frac {5 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}-\frac {5 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \text {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^{9/4} c^{3/4}}-\frac {5 \text {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^{9/4} c^{3/4}}\\ &=\frac {x^{3/2}}{4 b \left (b+c x^2\right )^2}+\frac {5 x^{3/2}}{16 b^2 \left (b+c x^2\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{9/4} c^{3/4}}+\frac {5 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}-\frac {5 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{9/4} c^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 138, normalized size = 0.58 \begin {gather*} \frac {\frac {4 \sqrt [4]{b} x^{3/2} \left (9 b+5 c x^2\right )}{\left (b+c x^2\right )^2}-\frac {5 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{c^{3/4}}-\frac {5 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{c^{3/4}}}{64 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 150, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}}}{4 b \left (c \,x^{2}+b \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{16 b \left (c \,x^{2}+b \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{b}\) | \(150\) |
default | \(\frac {x^{\frac {3}{2}}}{4 b \left (c \,x^{2}+b \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{16 b \left (c \,x^{2}+b \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{b}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 217, normalized size = 0.91 \begin {gather*} \frac {5 \, c x^{\frac {7}{2}} + 9 \, b x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )}} + \frac {5 \, {\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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 250, normalized size = 1.05 \begin {gather*} -\frac {20 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {-b^{5} c \sqrt {-\frac {1}{b^{9} c^{3}}} + x} b^{2} c \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {1}{4}} - b^{2} c \sqrt {x} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {1}{4}}\right ) - 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {1}{4}} \log \left (b^{7} c^{2} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {1}{4}} \log \left (-b^{7} c^{2} \left (-\frac {1}{b^{9} c^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 4 \, {\left (5 \, c x^{3} + 9 \, b x\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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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 = 4.49, size = 209, normalized size = 0.87 \begin {gather*} \frac {5 \, c x^{\frac {7}{2}} + 9 \, b x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} 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 )}{64 \, b^{3} c^{3}} + \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 )}{64 \, b^{3} c^{3}} - \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 )}{128 \, b^{3} c^{3}} + \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 )}{128 \, b^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 86, normalized size = 0.36 \begin {gather*} \frac {\frac {9\,x^{3/2}}{16\,b}+\frac {5\,c\,x^{7/2}}{16\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {5\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{9/4}\,c^{3/4}}-\frac {5\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{9/4}\,c^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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