Optimal. Leaf size=308 \[ \frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{7/8} c^{9/8}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )} \]
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Rubi [A] time = 0.25, 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 = {288, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{7/8} c^{9/8}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 211
Rule 212
Rule 214
Rule 288
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\left (a+c x^4\right )^2} \, dx &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}+\frac {\int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{8 c}\\ &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 c}\\ &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 \sqrt {-a} c}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 \sqrt {-a} c}\\ &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/4} c}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/4} c}-\frac {\operatorname {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)^{3/4} c}-\frac {\operatorname {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)^{3/4} c}\\ &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\operatorname {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)^{3/4} c^{5/4}}-\frac {\operatorname {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)^{3/4} c^{5/4}}+\frac {\operatorname {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)^{7/8} c^{9/8}}+\frac {\operatorname {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)^{7/8} c^{9/8}}\\ &=-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}+\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)^{7/8} c^{9/8}}-\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)^{7/8} c^{9/8}}-\frac {\operatorname {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)^{7/8} c^{9/8}}+\frac {\operatorname {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)^{7/8} c^{9/8}}\\ &=-\frac {\sqrt {x}}{4 c \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)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{7/8} c^{9/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{7/8} c^{9/8}}+\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)^{7/8} c^{9/8}}-\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)^{7/8} c^{9/8}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.18 \[ \frac {\sqrt {x} \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {c x^4}{a}\right )}{4 a c}-\frac {\sqrt {x}}{4 c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 545, normalized size = 1.77 \[ \frac {4 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{2} c^{2} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + x} a^{6} c^{8} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} a^{6} c^{8} \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}} + 1\right ) + 4 \, \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{2} c^{2} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + x} a^{6} c^{8} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} a^{6} c^{8} \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}} - 1\right ) + \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{2} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + x\right ) - \sqrt {2} {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{2} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} a c \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + x\right ) + 8 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{2} c^{2} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{4}} + x} a^{6} c^{8} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}} - a^{6} c^{8} \sqrt {x} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {7}{8}}\right ) + 2 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \log \left (a c \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 2 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} \log \left (-a c \left (-\frac {1}{a^{7} c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 16 \, \sqrt {x}}{64 \, {\left (c^{2} x^{4} + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 486, normalized size = 1.58 \[ \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {1}{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 c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\sqrt {x}}{4 \, {\left (c x^{4} + a\right )} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 47, normalized size = 0.15 \[ -\frac {\sqrt {x}}{4 \left (c \,x^{4}+a \right ) c}+\frac {\ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{32 c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{7}} \]
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 \[ \frac {x^{\frac {9}{2}}}{4 \, {\left (a c x^{4} + a^{2}\right )}} - \int \frac {x^{\frac {7}{2}}}{8 \, {\left (a c x^{4} + a^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 135, normalized size = 0.44 \[ -\frac {\sqrt {x}}{4\,c\,\left (c\,x^4+a\right )}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{7/8}\,c^{9/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/8}\,c^{9/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{9/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{9/8}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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