Optimal. Leaf size=329 \[ \frac {9 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}+\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.28, 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 = {288, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}+\frac {9 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}+\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2} \]
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^{15/2}}{\left (a+c x^4\right )^3} \, dx &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}+\frac {9 \int \frac {x^{7/2}}{\left (a+c x^4\right )^2} \, dx}{16 c}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}+\frac {9 \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{128 c^2}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 c^2}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 \sqrt {-a} c^2}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 \sqrt {-a} c^2}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{3/4} c^2}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{3/4} c^2}-\frac {9 \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)^{3/4} c^2}-\frac {9 \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)^{3/4} c^2}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}-\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \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)^{3/4} c^{9/4}}-\frac {9 \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)^{3/4} c^{9/4}}+\frac {9 \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)^{7/8} c^{17/8}}+\frac {9 \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)^{7/8} c^{17/8}}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}-\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}+\frac {9 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \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)^{7/8} c^{17/8}}+\frac {9 \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)^{7/8} c^{17/8}}\\ &=-\frac {x^{9/2}}{8 c \left (a+c x^4\right )^2}-\frac {9 \sqrt {x}}{64 c^2 \left (a+c x^4\right )}+\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}+\frac {9 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}-\frac {9 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{7/8} c^{17/8}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 66, normalized size = 0.20 \[ \frac {\sqrt {x} \left (9 \left (a+c x^4\right )^2 \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {c x^4}{a}\right )-a \left (9 a+17 c x^4\right )\right )}{64 a c^2 \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 672, normalized size = 2.04 \[ \frac {36 \, \sqrt {2} {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{2} c^{4} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{4}} + \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + x} a^{6} c^{15} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}} - \sqrt {2} a^{6} c^{15} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}} + 1\right ) + 36 \, \sqrt {2} {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{2} c^{4} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{4}} - \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + x} a^{6} c^{15} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}} - \sqrt {2} a^{6} c^{15} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}} - 1\right ) + 9 \, \sqrt {2} {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{4} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{4}} + \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + x\right ) - 9 \, \sqrt {2} {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{4} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{4}} - \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + x\right ) + 72 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{2} c^{4} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{4}} + x} a^{6} c^{15} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}} - a^{6} c^{15} \sqrt {x} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {7}{8}}\right ) + 18 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \log \left (a c^{2} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 18 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} \log \left (-a c^{2} \left (-\frac {1}{a^{7} c^{17}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 16 \, {\left (17 \, c x^{4} + 9 \, a\right )} \sqrt {x}}{1024 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 496, normalized size = 1.51 \[ \frac {9 \, \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 c^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \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 c^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \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 c^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, \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 c^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, \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 c^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {9 \, \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 c^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \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 c^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {9 \, \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 c^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {17 \, c x^{\frac {9}{2}} + 9 \, a \sqrt {x}}{64 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 59, normalized size = 0.18 \[ \frac {9 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 c^{3} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{7}}+\frac {-\frac {17 x^{\frac {9}{2}}}{64 c}-\frac {9 a \sqrt {x}}{64 c^{2}}}{\left (c \,x^{4}+a \right )^{2}} \]
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 \[ \frac {9 \, c x^{\frac {17}{2}} + a x^{\frac {9}{2}}}{64 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} - 9 \, \int \frac {x^{\frac {7}{2}}}{128 \, {\left (a c^{2} x^{4} + a^{2} c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 158, normalized size = 0.48 \[ -\frac {\frac {17\,x^{9/2}}{64\,c}+\frac {9\,a\,\sqrt {x}}{64\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {9\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{7/8}\,c^{17/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,9{}\mathrm {i}}{256\,{\left (-a\right )}^{7/8}\,c^{17/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {9}{512}-\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{17/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {9}{512}+\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,c^{17/8}} \]
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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