Optimal. Leaf size=329 \[ \frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {105 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]
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Rubi [A] time = 0.30, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {290, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {105 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 211
Rule 212
Rule 214
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^3} \, dx &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \int \frac {1}{\sqrt {x} \left (a+c x^4\right )^2} \, dx}{16 a}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {105 \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{128 a^2}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {105 \operatorname {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a^2}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}-\frac {105 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{5/2}}-\frac {105 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{5/2}}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}-\frac {105 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}-\frac {105 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}-\frac {105 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}-\frac {105 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{11/4}}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}-\frac {105 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}-\frac {105 \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 )}{512 (-a)^{11/4} \sqrt [4]{c}}+\frac {105 \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 )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \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 )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}-\frac {105 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}\\ &=\frac {\sqrt {x}}{8 a \left (a+c x^4\right )^2}+\frac {15 \sqrt {x}}{64 a^2 \left (a+c x^4\right )}+\frac {105 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{23/8} \sqrt [8]{c}}+\frac {105 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}-\frac {105 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{23/8} \sqrt [8]{c}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 62, normalized size = 0.19 \[ \frac {\sqrt {x} \left (105 \left (a+c x^4\right )^2 \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {c x^4}{a}\right )+a \left (23 a+15 c x^4\right )\right )}{64 a^3 \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 631, normalized size = 1.92 \[ \frac {420 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} + 1\right ) + 420 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - \sqrt {2} a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - 1\right ) + 105 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x\right ) - 105 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + x\right ) + 840 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{6} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{4}} + x} a^{20} c \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}} - a^{20} c \sqrt {x} \left (-\frac {1}{a^{23} c}\right )^{\frac {7}{8}}\right ) + 210 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (a^{3} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 210 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} \log \left (-a^{3} \left (-\frac {1}{a^{23} c}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 16 \, {\left (15 \, c x^{4} + 23 \, a\right )} \sqrt {x}}{1024 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 472, normalized size = 1.43 \[ \frac {105 \, \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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {105 \, \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 )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {105 \, \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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {105 \, \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 )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {15 \, c x^{\frac {9}{2}} + 23 \, a \sqrt {x}}{64 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 62, normalized size = 0.19 \[ \frac {105 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a^{2} c \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{7}}+\frac {\frac {15 c \,x^{\frac {9}{2}}}{64 a^{2}}+\frac {23 \sqrt {x}}{64 a}}{\left (c \,x^{4}+a \right )^{2}} \]
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 \[ -105 \, c \int \frac {x^{\frac {7}{2}}}{128 \, {\left (a^{3} c x^{4} + a^{4}\right )}}\,{d x} + \frac {105 \, c^{2} x^{\frac {17}{2}} + 225 \, a c x^{\frac {9}{2}} + 128 \, a^{2} \sqrt {x}}{64 \, {\left (a^{3} c^{2} x^{8} + 2 \, a^{4} c x^{4} + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 157, normalized size = 0.48 \[ \frac {\frac {23\,\sqrt {x}}{64\,a}+\frac {15\,c\,x^{9/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {105\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,105{}\mathrm {i}}{256\,{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {105}{512}-\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,c^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {105}{512}+\frac {105}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{23/8}\,c^{1/8}} \]
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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