Optimal. Leaf size=329 \[ \frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.31, 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, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 290
Rule 297
Rule 298
Rule 301
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 \int \frac {x^{5/2}}{\left (a+c x^4\right )^2} \, dx}{16 a}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \int \frac {x^{5/2}}{a+c x^4} \, dx}{128 a^2}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \operatorname {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a^2}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}}+\frac {9 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^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^2 c^{3/4}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a^2 c^{3/4}}-\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^2 c^{3/4}}+\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^2 c^{3/4}}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\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^2 c}+\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^2 c}+\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)^{17/8} c^{7/8}}+\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)^{17/8} c^{7/8}}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\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)^{17/8} c^{7/8}}-\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)^{17/8} c^{7/8}}+\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)^{17/8} c^{7/8}}-\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)^{17/8} c^{7/8}}\\ &=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^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)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\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)^{17/8} c^{7/8}}-\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)^{17/8} c^{7/8}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.09 \[ \frac {2 x^{7/2} \, _2F_1\left (\frac {7}{8},3;\frac {15}{8};-\frac {c x^4}{a}\right )}{7 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 672, normalized size = 2.04 \[ -\frac {36 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{15} c^{6} \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} - a^{13} c^{5} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {3}{4}} + x} a^{2} c \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} + 1\right ) + 36 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{15} c^{6} \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} - a^{13} c^{5} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {3}{4}} + x} a^{2} c \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} - \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} - 1\right ) - 9 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{15} c^{6} \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} - a^{13} c^{5} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {3}{4}} + x\right ) + 9 \, \sqrt {2} {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{15} c^{6} \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} - a^{13} c^{5} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {3}{4}} + x\right ) + 72 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {-a^{13} c^{5} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {3}{4}} + x} a^{2} c \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} - a^{2} c \sqrt {x} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}}\right ) - 18 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 18 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 16 \, {\left (9 \, c x^{7} + 17 \, a x^{3}\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 = 0.86, size = 472, normalized size = 1.43 \[ \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, \left (\frac {a}{c}\right )^{\frac {7}{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 {9 \, c x^{\frac {15}{2}} + 17 \, a x^{\frac {7}{2}}}{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 \[ \frac {9 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a^{2} c \RootOf \left (c \,\textit {\_Z}^{8}+a \right )}+\frac {\frac {9 c \,x^{\frac {15}{2}}}{64 a^{2}}+\frac {17 x^{\frac {7}{2}}}{64 a}}{\left (c \,x^{4}+a \right )^{2}} \]
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 \[ \frac {9 \, c x^{\frac {15}{2}} + 17 \, a x^{\frac {7}{2}}}{64 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 9 \, \int \frac {x^{\frac {5}{2}}}{128 \, {\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 157, normalized size = 0.48 \[ \frac {\frac {17\,x^{7/2}}{64\,a}+\frac {9\,c\,x^{15/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {9\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,9{}\mathrm {i}}{256\,{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {9}{512}-\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {9}{512}+\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{17/8}\,c^{7/8}} \]
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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