Optimal. Leaf size=299 \[ -\frac {(-a)^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac {2 x^{3/2}}{3 c} \]
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Rubi [A] time = 0.38, 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 = {321, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac {(-a)^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac {2 x^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 297
Rule 298
Rule 300
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{9/2}}{a+c x^4} \, dx &=\frac {2 x^{3/2}}{3 c}-\frac {a \int \frac {\sqrt {x}}{a+c x^4} \, dx}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x^2}{a+c x^8} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {\sqrt {-a} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}-\frac {\sqrt {-a} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {\sqrt {-a} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/4}}+\frac {\sqrt {-a} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/4}}+\frac {\sqrt {-a} \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 c^{5/4}}-\frac {\sqrt {-a} \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 c^{5/4}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {\sqrt {-a} \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 c^{3/2}}-\frac {\sqrt {-a} \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 c^{3/2}}-\frac {(-a)^{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} c^{11/8}}-\frac {(-a)^{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} c^{11/8}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{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} c^{11/8}}+\frac {(-a)^{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} c^{11/8}}-\frac {(-a)^{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} c^{11/8}}+\frac {(-a)^{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} c^{11/8}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{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} c^{11/8}}+\frac {(-a)^{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} c^{11/8}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.10 \[ -\frac {2 x^{3/2} \left (\, _2F_1\left (\frac {3}{8},1;\frac {11}{8};-\frac {c x^4}{a}\right )-1\right )}{3 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 491, normalized size = 1.64 \[ \frac {12 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - a^{3}}{a^{3}}\right ) + 12 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} + a^{3}}{a^{3}}\right ) - 3 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x\right ) + 3 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x\right ) - 24 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}}}{a^{3}}\right ) + 6 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{4} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a \sqrt {x}\right ) - 6 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (-c^{4} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a \sqrt {x}\right ) + 16 \, x^{\frac {3}{2}}}{24 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 453, normalized size = 1.52 \[ \frac {2 \, x^{\frac {3}{2}}}{3 \, c} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 \, c \sqrt {-2 \, \sqrt {2} + 4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 39, normalized size = 0.13 \[ \frac {2 x^{\frac {3}{2}}}{3 c}-\frac {a \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{4 c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{5}} \]
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 \[ -a \int \frac {\sqrt {x}}{c^{2} x^{4} + a c}\,{d x} + \frac {2 \, x^{\frac {3}{2}}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 126, normalized size = 0.42 \[ \frac {2\,x^{3/2}}{3\,c}+\frac {{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,c^{11/8}}+\frac {{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,c^{11/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{c^{11/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{c^{11/8}} \]
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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