Optimal. Leaf size=308 \[ \frac {x^{7/2}}{4 a \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}} \]
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Rubi [A]
time = 0.18, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {296, 335,
307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {x^{7/2}}{4 a \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 296
Rule 303
Rule 304
Rule 307
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (a+c x^4\right )^2} \, dx &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}+\frac {\int \frac {x^{5/2}}{a+c x^4} \, dx}{8 a}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 a \sqrt {c}}+\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 a \sqrt {c}}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 a c^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 a c^{3/4}}-\frac {\text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 a c^{3/4}}+\frac {\text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 a c^{3/4}}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 a c}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 a c}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}\\ &=\frac {x^{7/2}}{4 a \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{9/8} c^{7/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{9/8} c^{7/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{9/8} c^{7/8}}\\ \end {align*}
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Mathematica [A]
time = 1.14, size = 277, normalized size = 0.90 \begin {gather*} \frac {\frac {8 \sqrt [8]{a} x^{7/2}}{a+c x^4}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{7/8}}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{7/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{7/8}}-\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{7/8}}}{32 a^{9/8}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.14, size = 50, normalized size = 0.16
method | result | size |
derivativedivides | \(\frac {x^{\frac {7}{2}}}{4 a \left (x^{4} c +a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{32 a c}\) | \(50\) |
default | \(\frac {x^{\frac {7}{2}}}{4 a \left (x^{4} c +a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{32 a c}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs.
\(2 (207) = 414\).
time = 0.39, size = 542, normalized size = 1.76 \begin {gather*} \frac {16 \, x^{\frac {7}{2}} - 4 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{8} c^{6} \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} - a^{7} c^{5} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {3}{4}} + x} a c \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} + 1\right ) - 4 \, \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{8} c^{6} \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} - a^{7} c^{5} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {3}{4}} + x} a c \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} - 1\right ) + \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{8} c^{6} \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} - a^{7} c^{5} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {3}{4}} + x\right ) - \sqrt {2} {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{8} c^{6} \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} - a^{7} c^{5} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {3}{4}} + x\right ) - 8 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {-a^{7} c^{5} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {3}{4}} + x} a c \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} - a c \sqrt {x} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}}\right ) + 2 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \log \left (a^{8} c^{6} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 2 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {1}{8}} \log \left (-a^{8} c^{6} \left (-\frac {1}{a^{9} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right )}{64 \, {\left (a c x^{4} + a^{2}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs.
\(2 (207) = 414\).
time = 1.08, size = 462, normalized size = 1.50 \begin {gather*} \frac {x^{\frac {7}{2}}}{4 \, {\left (c x^{4} + a\right )} a} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 135, normalized size = 0.44 \begin {gather*} \frac {x^{7/2}}{4\,a\,\left (c\,x^4+a\right )}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{9/8}\,c^{7/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{9/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{7/8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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