Optimal. Leaf size=299 \[ -\frac {c^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {2}{3 a x^{3/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 299, 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 = {325, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac {c^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {2}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 211
Rule 212
Rule 301
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \left (a+c x^4\right )} \, dx &=-\frac {2}{3 a x^{3/2}}-\frac {c \int \frac {x^{3/2}}{a+c x^4} \, dx}{a}\\ &=-\frac {2}{3 a x^{3/2}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {x^4}{a+c x^8} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2}{3 a x^{3/2}}+\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2}{3 a x^{3/2}}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}+\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}+\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 (-a)^{5/4}}\\ &=-\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}+\frac {\sqrt [4]{c} \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 )}{4 (-a)^{5/4}}+\frac {\sqrt [4]{c} \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 )}{4 (-a)^{5/4}}-\frac {c^{3/8} \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 )}{4 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \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 )}{4 \sqrt {2} (-a)^{11/8}}\\ &=-\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}\\ &=-\frac {2}{3 a x^{3/2}}-\frac {c^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{11/8}}-\frac {c^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{11/8}}-\frac {c^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {c^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} (-a)^{11/8}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.10 \[ -\frac {2 \, _2F_1\left (-\frac {3}{8},1;\frac {5}{8};-\frac {c x^4}{a}\right )}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 553, normalized size = 1.85 \[ -\frac {12 \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{4} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}} - \sqrt {2} \sqrt {\sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} - a^{3} c^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + c^{4} x} a^{4} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}} - c^{3}}{c^{3}}\right ) + 12 \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{4} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}} - \sqrt {2} \sqrt {-\sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} - a^{3} c^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + c^{4} x} a^{4} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}} + c^{3}}{c^{3}}\right ) + 3 \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} - a^{3} c^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + c^{4} x\right ) - 3 \, \sqrt {2} a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{7} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} - a^{3} c^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + c^{4} x\right ) - 24 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a^{4} c^{2} \sqrt {x} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}} - \sqrt {-a^{3} c^{3} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{4}} + c^{4} x} a^{4} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {3}{8}}}{c^{3}}\right ) - 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 6 \, a x^{2} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-a^{7} \left (-\frac {c^{3}}{a^{11}}\right )^{\frac {5}{8}} + c^{2} \sqrt {x}\right ) + 16 \, \sqrt {x}}{24 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 461, normalized size = 1.54 \[ \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{2 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{2 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{4 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {c \left (\frac {a}{c}\right )^{\frac {5}{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 )}{4 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {2}{3 \, a x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 38, normalized size = 0.13 \[ -\frac {\ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{4 a \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{3}}-\frac {2}{3 a \,x^{\frac {3}{2}}} \]
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 \[ -c \int \frac {x^{\frac {3}{2}}}{a c x^{4} + a^{2}}\,{d x} - \frac {2}{3 \, a x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 126, normalized size = 0.42 \[ \frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}}{a^{1/8}}\right )}{2\,a^{11/8}}-\frac {2}{3\,a\,x^{3/2}}-\frac {{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{2\,a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-c\right )}^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{a^{11/8}} \]
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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